#499247

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
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×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

849 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)(7,10)(8,9)(11,12)(13,14)(15,26)(16,21)(17,23)(18,24)(19,33)(20,36)(22,30)(25,35)(27,40)(28,41)(29,38)(31,43)(32,37)(34,39)(42,44);(1,2)(3,4)(6,9)(10,12)(13,15)(14,21)(16,18)(17,29)(19,28)(20,27)(22,33)(23,40)(24,26)(25,39)(30,32)(31,34)(35,44)(36,38)(37,41)(42,43);(2,4)(13,16,18,15)(14,22,24,41)(17,29,20,27)(19,34,32,44)(21,37,26,33)(23,39,36,42)(25,40,43,38)(28,35,30,31);(5,7,8,11)(6,10,9,12)(13,17,18,20)(14,23,24,36)(15,27,16,29)(19,31,32,35)(21,38,26,40)(22,39,41,42)(25,33,43,37)(28,44,30,34)

Polycyclic presentation:

2^9|c^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406443);