#96217

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

512 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)(27,33,34,30)(28,32)(29,31)(35,39,40,37)(36,38);(2,6,8,7)(9,11)(10,14,16,17)(12,18)(13,19)(15,22,20,23)(21,24)(25,27,28,35,29,34,36,40)(26,30,32,37,31,33,38,39);(1,3,5,4)(2,6,8,7)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ghi,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, 3413);