#223350

order	512
nilpotency class	3
isoclinism	isoclinic class of #213564
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1179 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=eh,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ci,d^b=de,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 130546);