#7605739

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

1355 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)(9,10,13,12)(11,14,16,15)(17,18)(19,25)(20,34)(21,39)(22,28)(23,29)(24,30)(26,36)(27,33)(31,42)(32,41)(35,40)(37,45)(38,44)(43,46)(47,48);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,25)(20,35)(21,23)(22,32)(24,33)(26,42)(27,30)(28,41)(29,39)(31,36)(34,40)(37,47)(38,46)(43,44)(45,48);(1,3,5,4)(2,7,8,6)(9,12,13,10)(11,15,16,14)(17,20,23,37,24,38,22,36)(18,26,28,44,30,45,29,34)(19,31,32,46,33,47,21,35)(25,40,39,48,27,43,41,42);(1,3,5,4)(2,7,8,6)(11,16)(14,15)(17,21,22,33,24,32,23,19)(18,27,29,39,30,25,28,41)(20,35,36,47,38,46,37,31)(26,43,34,48,45,40,44,42);(1,4,5,3)(2,7,8,6)(17,22,24,23)(18,28,30,29)(19,32,33,21)(20,36,38,37)(25,41,27,39)(26,44,45,34)(31,46,47,35)(40,42,43,48)

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,c^2=fh,d^2=fg,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, 7512935);