#346290

order	512
nilpotency class	3
isoclinism	isoclinic class of #346284
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2840
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

494 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,8,28)(3,14,18,34)(4,20,29,37)(5,11,30,40)(6,12,7,13)(9,19,33,17)(10,22,53,23)(15,44,46,31)(16,43,24,55)(21,52,58,39)(25,56,54,36)(26,42,27,41)(32,51,38,50)(35,48,60,47)(45,61,49,59)(57,62,64,63);(1,3)(2,9,12,14)(4,15)(5,24,26,50)(6,17)(7,19)(8,18)(10,35,20,31)(11,38,40,32)(13,34,28,33)(16,27,51,30)(21,45,25,57)(22,47)(23,48)(29,46)(36,62,56,63)(37,44,53,60)(39,61,52,59)(41,55,42,43)(49,54,64,58);(1,4)(2,10,12,20)(3,15)(5,25,26,21)(6,23)(7,22)(8,29)(9,31,14,35)(11,39,40,52)(13,37,28,53)(16,49,51,64)(17,48)(18,46)(19,47)(24,45,50,57)(27,58,30,54)(32,61,38,59)(33,44,34,60)(36,41,56,42)(43,63,55,62);(1,5,6,27)(2,11,12,41)(3,16,17,50)(4,21,23,54)(7,26,8,30)(9,32,14,43)(10,36,20,52)(13,42,28,40)(15,45,48,64)(18,24,19,51)(22,25,29,58)(31,59,35,62)(33,38,34,55)(37,39,53,56)(44,61,60,63)(46,49,47,57)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,d^b=defh,e^b=eh,f^b=fhi,d^c=de,e^c=ei,f^c=fh,g^d=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72340967
Fitting subgroup of	#72340967
maximal subgroup of	#72235843, #72235846, #72340928, …
maximal normal subgroup of	#72235843, #72235846, #72340928, …
maximal quotient of	#72235749, #72235753

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253486);