#93254

order	512
nilpotency class	3
isoclinism	isoclinic class of #7333
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₄×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#322
commutator	C₈	C₈²
Frattini, cosocle	#1726	C₂²
socle	C₂	#7014
prime radical	C₂³	H(3,ℤ/4)
LCS	G ▷ C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ #1726 ◁ G
LECS	G ▷ #1726 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

191 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,f^2=i,b^a=bc,c^a=ci,e^a=efi,h^a=hi,c^b=ci,d^b=df,g^b=gi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#93254, #72210684, #72210685, …
maximal normal subgroup of	C₂×#93254, #72210684, #72210685, …
maximal quotient of	#72210248, #72210250, C₂×#93254

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 450);