#6459190

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1490 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)(11,14)(12,13)(15,16)(17,18,23,36,22,26,21,27)(19,28,35,24,31,43,30,39)(20,25,37,47,34,41,33,42)(29,40,46,38,44,48,32,45);(2,6)(7,8)(17,19,22,31)(18,24,26,39)(20,32,34,46)(21,35,23,30)(25,40,41,48)(27,43,36,28)(29,37,44,33)(38,47,45,42);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,23,37,22,34,21,33)(18,25,36,47,26,41,27,42)(19,29,35,46,31,44,30,32)(24,38,43,48,39,45,28,40);(1,4,5,3)(2,8,6,7)(17,20,21,33,22,34,23,37)(18,25,27,42,26,41,36,47)(19,29,30,32,31,44,35,46)(24,38,28,40,39,45,43,48);(9,11,12,15)(10,14,13,16)(18,26)(24,39)(25,41)(27,36)(28,43)(38,45)(40,48)(42,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=fi,d^2=f,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di

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, 6366386);