#7594421

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1439 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)(3,5)(4,6)(7,8)(17,18)(19,29)(20,25)(21,24)(22,37)(23,28)(26,31)(27,38)(30,42)(32,41)(33,47)(34,48)(35,39)(36,40)(43,46)(44,45);(1,3)(2,5)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19,22,21,23,31,38,36)(18,24,27,29,28,40,37,26)(20,30,34,35,32,45,46,33)(25,39,43,42,41,47,48,44);(2,6)(3,7)(9,11,12,15)(10,13,14,16)(17,20)(18,25)(19,30)(21,35)(22,34)(23,32)(24,39)(26,44)(27,43)(28,41)(29,42)(31,45)(33,36)(37,48)(38,46)(40,47);(2,6)(5,8)(10,14)(13,16)(20,32)(25,41)(30,45)(33,35)(34,46)(39,47)(42,44)(43,48);(17,21,22,31,23,36,38,19)(18,26,27,24,28,29,37,40)(20,33,34,30,32,35,46,45)(25,42,43,47,41,44,48,39)

Polycyclic presentation:

2^9|b^2=f,c^2=h,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,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, 7501617);