#7592757

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1291 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)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,32)(21,38)(22,27)(23,39)(24,28)(26,36)(29,43)(30,41)(31,42)(33,40)(34,47)(35,44)(37,46)(45,48);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,24,31)(18,25,28,42)(20,33,37,34)(21,22,30,23)(26,43,44,48)(27,41,39,38)(29,35,45,36)(32,40,46,47);(17,20,22,35,24,37,23,36)(18,26,39,46,28,44,27,32)(19,29,21,34,31,45,30,33)(25,40,41,48,42,47,38,43);(11,15)(13,16)(17,21,23,19,24,30,22,31)(18,25,27,38,28,42,39,41)(20,34,36,29,37,33,35,45)(26,40,32,43,44,47,46,48);(1,3,4,8)(2,6,5,7)(17,22,24,23)(18,27,28,39)(19,30,31,21)(20,35,37,36)(25,41,42,38)(26,32,44,46)(29,33,45,34)(40,48,47,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=gi,d^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,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, 7499953);