#457863

order	512
nilpotency class	3
isoclinism	isoclinic class of #453981
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1008 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)(5,6,8,10)(7,11,12,9)(14,15)(18,19)(22,26)(27,35)(28,33)(29,36);(13,14,17,15)(16,18,20,19)(21,22,24,26)(23,29,30,36)(25,33,32,28)(27,34,35,31);(5,7,8,12)(6,9,10,11)(13,15,17,14)(16,18,20,19)(21,23)(22,27)(24,30)(25,34)(26,35)(28,36)(29,33)(31,32);(2,4)(13,16,17,20)(14,18,15,19)(23,30)(27,35)(29,36)(31,34)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 365059);