#346261

order	512
nilpotency class	3
isoclinism	isoclinic class of #346224
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

460 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 92 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)(3,14,18,8)(4,20,24,34)(5,26,28,10)(6,13)(7,11)(9,23,33,25)(12,15)(16,39,43,59)(17,45,62,35)(19,38,40,31)(21,49,44,29)(22,51,54,57)(27,37,41,36)(30,46,56,47)(32,53,48,55)(42,63)(50,61)(52,60)(58,64);(1,3,15,40)(2,8,12,38)(4,21,24,16)(5,17,41,62)(6,19,7,18)(9,29,33,59)(10,35,37,45)(11,14,13,31)(20,39,34,49)(22,52,55,64)(23,44,25,43)(26,56,36,30)(27,46,28,47)(32,60,51,58)(42,53,50,54)(48,63,57,61);(1,4)(2,9,13,34)(3,16,18,44)(5,22)(6,23)(7,25)(8,29,38,59)(10,32,26,57)(11,20,12,33)(14,39,31,49)(15,24)(17,42,46,52)(19,43,40,21)(27,54)(28,53)(30,58,56,60)(35,61,45,63)(36,48,37,51)(41,55)(47,64,62,50);(1,5,7,27)(2,10,11,36)(3,17,18,46)(4,22,25,54)(6,28,15,41)(8,30,14,35)(9,32,20,48)(12,37,13,26)(16,42,44,52)(19,47,40,62)(21,50,43,64)(23,53,24,55)(29,58,39,61)(31,45,38,56)(33,51,34,57)(49,63,59,60)

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283912, #72284110, #72284527, …
maximal normal subgroup of	#72283912, #72284110, #72284527, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253457);