#95329

order	512
nilpotency class	3
isoclinism	isoclinic class of #95218
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

800 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gh,d^2=gi,e^2=gh,f^2=gh,b^a=bd,c^a=cgh,d^a=dg,f^a=fghi,c^b=cg,d^b=dh,e^b=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2525);