#7608204

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1260 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=ghi,d^2=gi,e^2=fgi,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, 7515400);