#466565

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

936 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=efg,c^2=e,d^2=efi,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, 373761);