#6519344

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500450
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=fi,d^2=hi,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=dh,e^d=ei

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, 6426540);