#444310

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1187 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:

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

Polycyclic presentation:

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

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, 351506);