#6519355

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1436 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)(19,26)(20,25)(21,27)(22,28)(23,29)(24,30)(31,42)(32,41)(33,40)(34,39)(35,48)(36,44)(37,45)(38,46)(43,47);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,32,24,33,22,20)(18,25,28,40,30,41,29,26)(21,31,36,35,38,47,37,34)(27,39,45,43,46,48,44,42);(1,3,4,7)(2,6,5,8)(10,13)(11,15)(17,20,22,33,24,32,23,19)(18,26,29,41,30,40,28,25)(21,35,37,31,38,34,36,47)(27,43,44,39,46,42,45,48);(9,11,12,15)(10,14,13,16)(17,21)(18,27)(19,31)(20,34)(22,37)(23,36)(24,38)(25,39)(26,42)(28,45)(29,44)(30,46)(32,35)(33,47)(40,43)(41,48);(10,13)(14,16)(17,22,24,23)(18,28,30,29)(19,20,33,32)(21,36,38,37)(25,40,41,26)(27,44,46,45)(31,35,47,34)(39,42,48,43)

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,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, 6426551);