#7586827

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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1500 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)(3,8)(4,6)(5,7)(10,14)(11,12)(17,18)(19,28)(20,33)(21,37)(22,27)(23,26)(24,32)(25,36)(29,40)(30,41)(31,42)(34,44)(35,45)(38,46)(39,43)(47,48);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,16,12,15)(17,19)(18,24)(20,34)(21,31)(22,32)(23,39)(25,44)(26,42)(27,28)(29,36)(30,35)(33,40)(37,43)(38,48)(41,46)(45,47);(1,4,5,3)(2,6,7,8)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,29)(21,38)(22,36)(23,35)(24,40)(26,46)(27,33)(28,44)(30,31)(32,34)(37,45)(39,48)(41,43)(42,47);(9,11,13,12)(10,16,14,15)(17,20)(18,25)(19,30)(21,35)(22,36)(23,38)(24,41)(26,45)(27,33)(28,47)(29,31)(32,48)(34,39)(37,46)(40,43)(42,44);(2,7)(6,8)(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,26,27,37)(19,31,32,39)(20,35,36,38)(24,42,28,43)(25,45,33,46)(29,48,34,30)(40,41,44,47)

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,d^2=gi,e^2=g,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, 7494023);