#430540

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1068 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)(5,6,9,8)(7,11,12,10)(13,14)(15,19)(16,18)(17,20)(21,22)(23,28)(24,29)(25,27)(26,30)(31,36)(32,34)(33,35);(5,7,9,12)(6,10,8,11)(13,15)(14,17)(16,20)(18,19)(22,27)(23,31)(24,26)(28,35)(32,33)(34,36);(14,18)(17,19)(21,23,25,32)(22,28,27,34)(24,31,26,33)(29,36,30,35);(2,4)(5,8,9,6)(7,10,12,11)(15,20)(17,19)(21,24,25,26)(22,29,27,30)(23,31,32,33)(28,36,34,35)

Polycyclic presentation:

2^9|a^2=efi,b^2=efi,c^2=i,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,g^c=gi

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, 337736);