#93615

order	512
nilpotency class	3
isoclinism	isoclinic class of #93508
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,ℤ/4)
commutator	C₂²×C₈	C₄²
Frattini, cosocle	#1726	C₂²
socle	C₂²	#1290
prime radical	C₂³	#341
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ #1726 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

169 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 72 points:

(1,2,4,7,5,8,3,6)(9,10,13,20,16,23,14,42,17,24,41,57,15,22,40,21)(11,25,27,59,30,50,28,60,31,62,68,71,29,61,67,72)(12,32,34,52,37,19,35,51,38,53,39,58,36,54,43,56)(18,44,45,65,48,69,46,26,49,63,55,33,47,64,70,66);(2,6,8,7)(9,11,14,28,16,30,41,68,17,31,40,67,15,29,13,27)(10,18,21,46,23,48,20,55,24,49,42,70,22,47,57,45)(12,33,35,44,37,66,39,69,38,65,43,63,36,26,34,64)(19,50,52,71,53,62,51,72,54,61,58,59,32,25,56,60)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=fi,e^2=g,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efghi,h^a=hi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 811);