#473894

order	512
nilpotency class	3
isoclinism	isoclinic class of #453960
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

1124 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=eghi,c^2=i,d^2=e,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, 381090);