#7614865

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1483 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)(9,10,12,14)(11,15,16,13)(17,18)(19,29)(20,33)(21,38)(22,27)(23,28)(24,31)(25,36)(26,40)(30,41)(32,46)(34,43)(35,42)(37,45)(39,47)(44,48);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,27,29)(20,34,36,30)(21,37,40,48)(23,35,39,32)(25,43,33,41)(26,45,38,44)(28,46,47,42);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20,22,36)(18,25,27,33)(19,30,31,34)(21,32,40,35)(23,37,39,48)(24,41,29,43)(26,46,38,42)(28,44,47,45);(10,14)(13,15)(17,21,39,34,22,40,23,30)(18,26,47,41,27,38,28,43)(19,20,35,48,31,36,32,37)(24,33,46,44,29,25,42,45);(2,5)(6,8)(10,14)(13,15)(18,27)(20,36)(21,40)(24,29)(28,47)(30,34)(37,48)(42,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fgi,d^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,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, 7522061);