#346255

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

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

normal subgroups only

This group has 456 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)(3,14,17,8)(4,19,49,34)(5,23,26,10)(6,11)(7,13)(9,21,33,22)(12,27)(15,39,41,31)(16,44,46,35)(18,38,48,30)(20,52,55,57)(24,36,25,37)(28,42,50,43)(29,45,56,47)(32,54,51,58)(40,62)(53,59)(60,64)(61,63);(1,3,7,18)(2,8,13,30)(4,15,21,41)(5,16,26,47)(6,17,27,48)(9,31,34,39)(10,35,23,56)(11,14,12,38)(19,50,33,28)(20,53,54,40)(22,43,49,42)(24,46,25,45)(29,37,44,36)(32,59,57,62)(51,61,52,60)(55,63,58,64);(1,4,6,22)(2,9,12,33)(3,15)(5,20,25,55)(7,21,27,49)(8,28,30,50)(10,32,37,52)(11,19,13,34)(14,39,38,31)(16,40)(17,43)(18,41)(23,57,36,51)(24,58,26,54)(29,59,44,62)(35,61,56,60)(42,48)(45,64)(46,63)(47,53);(1,5,6,25)(2,10,11,36)(3,16,17,45)(4,20,22,55)(7,26,27,24)(8,29,14,35)(9,32,19,51)(12,37,13,23)(15,40,43,64)(18,47,48,46)(21,54,49,58)(28,59,39,61)(30,44,38,56)(31,60,50,62)(33,52,34,57)(41,53,42,63)

Polycyclic presentation:

2^9|a^2=fgh,b^2=i,c^2=efh,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	#72283608, #72283815, #72283882, …
maximal normal subgroup of	#72283608, #72283815, #72283882, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253451);