#7615379

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1399 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=g,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,e^c=ei,e^d=ei

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, 7522575);