#96148

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	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

568 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=hi,d^2=gi,e^2=i,f^2=i,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, 3344);