#7608112

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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

1258 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)(30,46)(32,41)(33,42)(34,40)(35,45)(36,44)(38,48)(43,47);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,15)(14,16)(17,19,23,31,24,32,21,22)(18,25,27,28,29,37,39,41)(20,30,33,34,35,47,36,38)(26,40,44,46,45,48,42,43);(1,4,5,2)(3,7,8,6)(10,15)(11,16)(17,20,24,35)(18,26,29,45)(19,30,32,47)(21,36,23,33)(22,38,31,34)(25,40,37,48)(27,44,39,42)(28,46,41,43);(3,8)(6,7)(9,12,14,13)(10,11,15,16)(17,21,24,23)(18,27,29,39)(19,31,32,22)(20,33,35,36)(25,41,37,28)(26,42,45,44)(30,38,47,34)(40,46,48,43);(9,13,14,12)(10,11,15,16)(17,22,21,32,24,31,23,19)(18,28,39,25,29,41,27,37)(20,34,36,30,35,38,33,47)(26,43,42,48,45,46,44,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=gh,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,d^c=di,e^c=ei,e^d=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, 7515308);