#6519504

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500416
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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1546 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=f,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,e^b=eh,d^c=dh,e^c=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, 6426700);