#475802

order	512
nilpotency class	3
isoclinism	isoclinic class of #457092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,14,18,23)(15,25,27,21)(16,29,32,22)(17,34,28,42)(19,24,20,36)(26,37,43,39)(30,38,31,44)(33,41,35,40);(6,9)(10,12)(13,15,19,28,18,27,20,17)(14,21,24,34,23,25,36,42)(16,30,33,43,32,31,35,26)(22,38,40,39,29,44,41,37);(2,4)(5,7)(6,10)(8,11)(9,12)(13,16)(14,22)(15,26)(17,30)(18,32)(19,35)(20,33)(21,37)(23,29)(24,41)(25,39)(27,43)(28,31)(34,44)(36,40)(38,42);(7,11)(10,12)(13,17,20,27,18,28,19,15)(14,21,24,34,23,25,36,42)(16,31,33,26,32,30,35,43)(22,39,41,38,29,37,40,44)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 382998);