#362985

order	512
nilpotency class	3
isoclinism	isoclinic class of #188176
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1392 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gi,d^2=f,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270181);