#6519385

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fi,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,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, 6426581);