#93595

order	512
nilpotency class	3
isoclinism	isoclinic class of #93508
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

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

165 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=g,e^2=f,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, 791);