#7492885

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324564
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	#45387	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1811 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)(3,6)(4,5)(7,8)(17,18)(19,24)(20,30)(21,26)(22,37)(23,27)(25,32)(28,40)(29,38)(31,45)(33,46)(34,47)(35,42)(36,48)(39,44)(41,43);(2,5)(3,7)(10,14)(11,12)(17,19,23,22)(18,24,27,37)(20,28,33,32)(21,29,35,36)(25,30,40,46)(26,38,42,48)(31,43,34,44)(39,45,41,47);(9,10)(11,16)(12,15)(13,14)(17,20,22,32,23,33,19,28)(18,25,24,30,27,40,37,46)(21,34,36,43,35,31,29,44)(26,41,38,47,42,39,48,45);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,21,23,35)(18,26,27,42)(19,29,22,36)(20,31,33,34)(24,38,37,48)(25,39,40,41)(28,43,32,44)(30,45,46,47);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(21,35)(26,42)(29,36)(31,34)(38,48)(39,41)(43,44)(45,47)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dhi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7400081);