#7624489

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624184
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 ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1264 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=f,e^2=i,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,e^b=ei,g^c=gh,h^c=hi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531685);