#7617084

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1317 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,32)(21,38)(22,27)(23,28)(25,36)(26,30)(29,42)(31,41)(33,40)(34,43)(35,45)(37,44)(39,46)(47,48);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,33)(21,39)(22,30)(23,31)(25,42)(26,27)(28,41)(29,36)(32,40)(34,37)(35,47)(38,46)(43,44)(45,48);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,23,36)(18,25,28,32)(19,29,31,33)(21,34,30,47)(22,37,39,35)(24,40,41,42)(26,43,38,48)(27,45,46,44);(1,4,5,3)(2,8,6,7)(10,14)(11,16)(17,21,39,19,23,30,22,31)(18,26,46,41,28,38,27,24)(20,34,35,29,36,47,37,33)(25,43,44,42,32,48,45,40);(2,6)(7,8)(10,14)(13,15)(17,22,23,39)(18,27,28,46)(19,30,31,21)(20,35,36,37)(24,26,41,38)(25,44,32,45)(29,34,33,47)(40,48,42,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=fg,d^2=fhi,e^2=i,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,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, 7524280);