#382317

order	512
nilpotency class	3
isoclinism	isoclinic class of #29742
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

774 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)(11,16)(12,14)(13,15)(17,18,22,26,21,25,23,20)(19,27,32,28,30,24,31,29);(1,2,4,5)(3,6,7,8)(17,18,23,20,21,25,22,26)(19,28,32,24,30,29,31,27);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19,21,30)(18,24,25,27)(20,28,26,29)(22,32,23,31);(3,7)(6,8)(9,12,13,11)(10,14,15,16)(17,20,23,25,21,26,22,18)(19,29,31,24,30,28,32,27)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=f,d^2=fh,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,d^c=de,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289513);