#94322

order	512
nilpotency class	5
isoclinism	isoclinic class of #94303
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1393
commutator	C₂×M₁₆	C₄²
Frattini, cosocle	C₂×#408	C₂²
socle	C₂	#7250
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂×#408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

344 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 60 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 — 2 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=i,e^2=i,f^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,h^a=hi,c^b=cg,d^b=di,f^b=fi,g^b=gh,h^b=hi,e^c=eh,g^c=gi,g^e=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1518);