#382128

order	512
nilpotency class	3
isoclinism	isoclinic class of #29592
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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 ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

966 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)(11,16)(12,14)(13,15)(17,18,24,28,22,27,23,21)(19,29,30,20,26,25,31,32);(2,5)(6,7)(17,19)(18,25)(20,21)(22,26)(23,31)(24,30)(27,29)(28,32);(1,3)(2,6)(4,8)(5,7)(9,11,13,12)(10,14,15,16)(17,20,22,32)(18,26,27,19)(21,30,28,31)(23,29,24,25);(3,8)(6,7)(9,12,13,11)(10,14,15,16)(17,21,23,27,22,28,24,18)(19,20,31,29,26,32,30,25)

Polycyclic presentation:

2^9|a^2=h,c^2=ef,d^2=efh,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gi,d^c=de,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, 289324);