#7608207

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

1265 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)(17,18,24,29)(19,30,33,28)(20,26,36,45)(21,27,23,39)(22,25,32,41)(31,43,47,46)(34,42,37,44)(35,48,38,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,15)(14,16)(17,19,23,22,24,33,21,32)(18,25,27,30,29,41,39,28)(20,31,34,38,36,47,37,35)(26,40,44,43,45,48,42,46);(1,4,5,2)(3,7,8,6)(10,15)(11,16)(17,20)(18,26)(19,31)(21,37)(22,38)(23,34)(24,36)(25,40)(27,44)(28,46)(29,45)(30,43)(32,35)(33,47)(39,42)(41,48);(3,8)(6,7)(9,12,14,13)(10,11,15,16)(17,21,24,23)(18,27,29,39)(19,32,33,22)(20,34,36,37)(25,30,41,28)(26,42,45,44)(31,38,47,35)(40,46,48,43);(9,13,14,12)(10,11,15,16)(17,22,23,33,24,32,21,19)(18,28,27,25,29,30,39,41)(20,35,34,31,36,38,37,47)(26,43,44,48,45,46,42,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=gh,d^2=gi,e^2=fg,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, 7515403);