#96160

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

580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2)(3,5)(4,7)(6,8)(10,14)(11,12)(15,16)(17,18,20,19,21,24,22,23);(2,5,7,6)(9,10)(11,14)(12,16)(13,15)(17,19,22,24,21,23,20,18);(9,11,13,12)(10,14,15,16)(17,20,21,22)(18,23,24,19)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=gi,e^2=i,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, 3356);