#6489288

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1519 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,18)(19,29)(20,34)(21,26)(22,27)(23,38)(24,28)(25,32)(30,45)(31,41)(33,46)(35,43)(36,44)(37,42)(39,48)(40,47);(11,12)(14,15)(17,19,23,32,24,33,22,31)(18,25,27,29,28,41,38,46)(20,30,35,47,21,36,37,48)(26,39,43,44,34,40,42,45);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,16,15)(17,20,24,21)(18,26,28,34)(19,30,33,36)(22,37,23,35)(25,39,41,40)(27,43,38,42)(29,44,46,45)(31,48,32,47);(9,12,13,11)(10,15,16,14)(17,21,24,20)(18,26,28,34)(19,30,33,36)(22,35,23,37)(25,40,41,39)(27,43,38,42)(29,45,46,44)(31,48,32,47);(1,4,5,3)(2,6,7,8)(10,16)(14,15)(17,22,24,23)(18,27,28,38)(19,31,33,32)(20,35,21,37)(25,29,41,46)(26,42,34,43)(30,47,36,48)(39,45,40,44)

Polycyclic presentation:

2^9|b^2=f,c^2=gh,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6396484);