#7617558

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1205 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=fg,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7524754);