#95219

order	512
nilpotency class	3
isoclinism	isoclinic class of #95218
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

944 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)(11,15)(12,14)(13,16)(17,18)(19,25,27,22)(20,28)(21,24)(23,30)(26,31,29,32);(1,3)(2,5,6,7)(4,8)(10,14)(11,12)(15,16)(17,19)(18,22)(20,29)(21,27)(23,32)(24,25)(26,30)(28,31);(2,6)(5,7)(9,11,13,12)(10,15,16,14)(17,20,21,30)(18,23,24,28)(19,26,27,29)(22,31,25,32)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=gi,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, 2415);