#7624408

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 ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1186 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(18,24)(19,28)(20,23)(22,34)(25,39)(27,44)(29,47)(30,46)(32,33)(35,36)(37,41)(38,43)(40,48)(42,45);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,18,21,24)(19,25,30,37)(20,31,23,26)(22,27,33,38)(28,41,46,39)(29,42,36,40)(32,44,34,43)(35,45,47,48);(17,19)(18,25)(20,29)(21,30)(22,35)(23,36)(24,37)(26,40)(27,45)(28,34)(31,42)(32,46)(33,47)(38,48)(39,44)(41,43);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,24)(25,37)(26,31)(27,38)(39,41)(40,42)(43,44)(45,48);(2,8)(6,9)(7,16)(14,15)(17,20,21,23)(18,26,24,31)(19,29,30,36)(22,32,33,34)(25,40,37,42)(27,43,38,44)(28,35,46,47)(39,45,41,48)

Polycyclic presentation:

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

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, 7531604);