#443247

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

976 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)(7,11)(13,14,16,18)(15,17,19,20)(21,22,27,32)(23,33,35,30)(24,31,26,29)(25,28,34,36);(5,6,8,9)(7,10,11,12)(13,15,16,19)(14,17,18,20)(21,23,27,35)(22,28,32,36)(24,25,26,34)(29,33,31,30);(5,7,8,11)(6,10,9,12)(15,19)(17,20)(21,24,27,26)(22,29,32,31)(23,34,35,25)(28,30,36,33);(2,4)(6,9)(10,12)(21,25,26,35,27,34,24,23)(22,30,31,28,32,33,29,36)

Polycyclic presentation:

2^9|a^2=g,b^2=fgi,c^2=f,d^2=efi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 350443);