#7590183

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,d^2=i,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,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, 7497379);