#95910

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	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=gi,e^2=g,f^2=g,b^a=bd,c^a=ch,d^a=dg,f^a=fghi,c^b=cg,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, 3106);