#7624713

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624235
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1218 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,8)(3,9,10,6)(4,12,13,7)(11,14,16,15)(17,18,21,26)(19,28,23,24)(20,31,32,47)(22,35,34,40)(25,41,42,33)(27,30,43,37)(29,46,38,44)(36,45,48,39);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19,21,23)(18,24,26,28)(20,29,32,38)(22,30,34,37)(25,39,42,45)(27,40,43,35)(31,44,47,46)(33,36,41,48);(17,20,21,32)(18,25,26,42)(19,29,23,38)(22,36,34,48)(24,39,28,45)(27,44,43,46)(30,41,37,33)(31,40,47,35);(3,10)(6,9)(11,16)(14,15)(18,26)(24,28)(25,42)(27,43)(31,47)(35,40)(39,45)(44,46);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(18,26)(24,28)(25,42)(27,43)(31,47)(35,40)(39,45)(44,46)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,g^c=gh,h^c=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531909);