#7593118

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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

1354 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)(10,14)(11,12)(17,18)(19,30)(20,34)(21,39)(22,29)(23,28)(24,27)(25,32)(26,38)(31,43)(33,42)(35,41)(36,46)(37,45)(40,44)(47,48);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(17,19)(18,25)(20,35)(21,22)(23,32)(24,33)(26,43)(27,39)(28,30)(29,42)(31,36)(34,44)(37,40)(38,47)(41,45)(46,48);(9,11,13,12)(10,15,14,16)(17,20,24,38,23,37,22,36)(18,26,29,34,28,46,27,45)(19,31,21,40,32,47,33,35)(25,41,39,48,30,44,42,43);(9,11,13,12)(10,16,14,15)(17,21,24,32,23,33,22,19)(18,25,29,39,28,30,27,42)(20,35,38,31,37,40,36,47)(26,44,34,43,46,41,45,48);(1,4,5,3)(2,7,8,6)(9,12,13,11)(10,15,14,16)(17,22,23,24)(18,27,28,29)(19,21,32,33)(20,36,37,38)(25,39,30,42)(26,45,46,34)(31,40,47,35)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=fghi,b^2=fi,c^2=g,d^2=g,e^2=f,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=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7500314);