#96081

order	512
nilpotency class	3
isoclinism	isoclinic class of #95192
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₄³	C₂³
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

(2,6,7,8)(9,10,12,16,14,18,13,17)(11,19,20,15,22,23,21,24)(25,26,29,32,30,33,28,37)(27,34,36,31,35,40,38,39);(1,2,4,6,5,7,3,8)(9,11,13,21,14,22,12,20)(10,15,16,23,18,24,17,19)(25,27)(26,31)(28,38)(29,36)(30,35)(32,40)(33,39)(34,37);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,24,19)(27,35)(31,39)(34,40)(36,38)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gh,d^2=gi,e^2=g,f^2=gh,b^a=bd,d^a=dg,f^a=fghi,c^b=ch,d^b=dh,e^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3277);