#500137

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3285	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1019 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ei,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=di,f^c=fh,g^c=gh,h^c=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, 407333);