#508227

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
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

768 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415423);