#7624354

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
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 ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1245 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(18,25)(19,28)(21,24)(23,36)(26,41)(27,44)(29,46)(30,48)(31,47)(33,40)(34,35)(37,38)(39,42)(43,45);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,18,22,25)(19,26,31,39)(20,21,32,24)(23,27,35,40)(28,42,47,41)(29,30,43,38)(33,34,44,36)(37,45,48,46);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19,22,31)(18,26,25,39)(20,29,32,43)(21,30,24,38)(23,37,35,48)(27,45,40,46)(28,34,47,36)(33,42,44,41);(3,11)(6,9)(10,16)(13,15)(17,20)(18,21)(19,29)(22,32)(23,33)(24,25)(26,30)(27,34)(28,46)(31,43)(35,44)(36,40)(37,42)(38,39)(41,48)(45,47);(2,8)(6,9)(7,14)(13,15)(17,21,22,24)(18,20,25,32)(19,30,31,38)(23,34,35,36)(26,29,39,43)(27,33,40,44)(28,37,47,48)(41,45,42,46)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531550);