#468851

order	512
nilpotency class	3
isoclinism	isoclinic class of #453960
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×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1228 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)(15,19)(17,20)(21,22,25,28,24,26,32,35)(23,29,31,34,30,36,33,27);(5,6,8,9)(7,10,11,12)(13,14,16,18)(15,20,19,17)(22,26)(23,30)(28,35)(31,33);(7,11)(10,12)(13,15,16,19)(14,17,18,20)(21,23,24,30)(22,27,26,34)(25,33,32,31)(28,36,35,29);(2,4)(5,7,8,11)(6,10,9,12)(23,30)(27,34)(29,36)(31,33)

Polycyclic presentation:

2^9|a^2=fgi,b^2=eg,c^2=g,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 376047);