#362590

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

940 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,19,21,25,22,26)(20,28,30,32,29,24,31,27);(2,5)(6,7)(9,11,12,16)(10,13,14,15)(17,19,23,25,21,26,22,18)(20,28,31,27,29,24,30,32);(10,14)(13,15)(17,20)(18,24)(19,27)(21,29)(22,31)(23,30)(25,28)(26,32);(1,3,4,8)(2,6,5,7)(10,14)(13,15)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,g^c=gi

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, 269786);