#302064

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

942 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)(9,10,13,14)(11,12,15,16)(17,18,22,25)(19,26,29,32)(20,24,21,30)(23,27,31,28);(1,3,5,4)(2,6,8,7)(11,15)(12,16);(9,11,12,14,13,15,16,10)(17,19,21,28,22,29,20,27)(18,23,24,26,25,31,30,32);(1,4,5,3)(2,7,8,6)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,26,31,32)

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=f,d^2=ehi,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, 209260);