#7582310

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1523 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=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, 7489506);