#304202

order	512
nilpotency class	3
isoclinism	isoclinic class of #267659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1086 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=ef,d^2=ei,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 211398);