#466962

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

970 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)(9,10,12,13)(11,14,15,16)(17,18,21,24)(19,27,30,23)(20,31,29,32)(22,25,28,26);(2,5)(6,8)(10,13)(14,16)(17,19,28,32,21,30,22,31)(18,23,26,29,24,27,25,20);(1,3,4,7)(2,6,5,8)(10,13)(11,15)(17,18)(19,29)(20,30)(21,24)(22,26)(23,32)(25,28)(27,31);(3,7)(6,8)(9,11,12,15)(10,14,13,16)(17,20)(18,19)(21,29)(22,27)(23,28)(24,30)(25,31)(26,32)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=ef,d^2=fi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 374158);