#444312

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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1155 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=fi,c^2=g,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, 351508);