#96216

order	512
nilpotency class	3
isoclinism	isoclinic class of #95196
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

516 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3412);