#467065

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1164 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(6,9)(10,12)(13,14,16,18)(15,17,19,20)(21,22,27,33,26,24,25,29)(23,28,34,36,30,32,31,35);(5,6,8,9)(7,10,11,12)(13,15,16,19)(14,17,18,20)(21,23)(22,28)(24,32)(25,31)(26,30)(27,34)(29,35)(33,36);(7,11)(10,12)(15,19)(17,20)(21,24,26,22)(23,28,30,32)(25,29,27,33)(31,36,34,35);(2,4)(5,7,8,11)(6,10,9,12)(23,30)(28,32)(31,34)(35,36)

Polycyclic presentation:

2^9|a^2=fg,b^2=egi,c^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,f^c=fi

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, 374261);