#7590769

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

1227 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fgi,c^2=gi,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,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, 7497965);