#6483087

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1395 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)(11,15)(12,16)(13,14)(17,18,24,29)(19,26,32,41)(20,33,31,25)(21,27,36,44)(22,28,23,38)(30,40,46,48)(34,47,45,39)(35,42,37,43);(2,5)(6,8)(10,14)(11,12)(17,19,23,31,24,32,22,20)(18,25,28,26,29,33,38,41)(21,30,35,45,36,46,37,34)(27,39,43,40,44,47,42,48);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20,22,32,24,31,23,19)(18,26,38,25,29,41,28,33)(21,34,37,46,36,45,35,30)(27,40,42,39,44,48,43,47);(9,12,13,11)(10,16,14,15)(17,21,24,36)(18,27,29,44)(19,30,32,46)(20,34,31,45)(22,37,23,35)(25,39,33,47)(26,40,41,48)(28,43,38,42);(10,14)(15,16)(17,22,24,23)(18,28,29,38)(19,20,32,31)(21,35,36,37)(25,26,33,41)(27,42,44,43)(30,45,46,34)(39,48,47,40)

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,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,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, 6390283);