#93255

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₈	C₈²
Frattini, cosocle	#1726	C₂²
socle	C₂	#7014
prime radical	C₂³	#338
LCS	G ▷ C₈ ▷ C₁
UCS	C₁ ◁ C₈ ◁ G
LECS	G ▷ #1726 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

183 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bc,e^a=efi,h^a=hi,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	#72210255, C₂×#93255, #72210650, …
maximal normal subgroup of	#72210255, C₂×#93255, #72210650, …
maximal quotient of	#72210248, C₂×#93255, #72210635

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 451);