#7539232

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1510 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)(9,10,12,14)(11,15,16,13)(17,18,22,27)(19,24,32,42)(20,33,36,25)(21,26,23,28)(29,43,34,39)(30,48,47,40)(31,41,38,46)(35,45,37,44);(1,3,5,4)(2,6,8,7)(11,16)(13,15)(17,19)(18,24)(20,34)(21,31)(22,32)(23,38)(25,43)(26,41)(27,42)(28,46)(29,36)(30,35)(33,39)(37,47)(40,44)(45,48);(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,37)(22,36)(23,35)(24,39)(26,45)(27,33)(28,44)(30,31)(32,34)(38,47)(40,41)(42,43)(46,48);(1,3,5,4)(2,7,8,6)(10,14)(13,15)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,40)(26,44)(27,33)(28,45)(29,31)(32,47)(34,38)(39,41)(42,48)(43,46);(1,4,5,3)(2,7,8,6)(17,21,22,23)(18,26,27,28)(19,31,32,38)(20,35,36,37)(24,41,42,46)(25,44,33,45)(29,47,34,30)(39,48,43,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=gi,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7446428);