#93614

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=f,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, 810);