#95413

order	512
nilpotency class	3
isoclinism	isoclinic class of #95192
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₄³	C₂³
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gh,d^2=gi,e^2=i,f^2=ghi,b^a=bd,c^a=cg,d^a=dg,f^a=fghi,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, 2609);