#7590190

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1445 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)(9,10,12,14)(11,13,15,16)(17,18)(19,28)(20,26)(21,27)(22,37)(23,38)(24,29)(25,31)(30,41)(32,44)(33,40)(34,47)(35,43)(36,42)(39,45)(46,48);(1,3)(2,5)(4,7)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,31,24,32,21,22)(18,25,27,28,29,37,38,44)(20,30,33,45,35,46,36,34)(26,39,42,41,43,47,40,48);(2,6)(3,7)(17,20)(18,26)(19,30)(21,36)(22,34)(23,33)(24,35)(25,39)(27,42)(28,41)(29,43)(31,45)(32,46)(37,47)(38,40)(44,48);(2,6)(5,8)(11,15)(13,16)(17,21,24,23)(18,27,29,38)(19,22,32,31)(20,33,35,36)(25,28,37,44)(26,40,43,42)(30,45,46,34)(39,48,47,41);(17,22,21,32,24,31,23,19)(18,28,38,25,29,44,27,37)(20,34,36,46,35,45,33,30)(26,41,40,39,43,48,42,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7497386);