#6463936

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1700 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)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18,23,27)(19,28,30,45)(20,33,36,25)(21,26,22,38)(24,31,40,32)(29,46,34,44)(35,43,37,42)(39,47,41,48);(2,5)(6,7)(10,14)(11,12)(17,19,21,31,23,30,22,32)(18,24,38,45,27,40,26,28)(20,34,37,47,36,29,35,48)(25,41,42,44,33,39,43,46);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,15,14,16)(17,20,23,36)(18,25,27,33)(19,29,30,34)(21,37,22,35)(24,39,40,41)(26,43,38,42)(28,44,45,46)(31,48,32,47);(9,12,13,11)(10,15,14,16)(19,30)(24,40)(28,45)(29,34)(31,32)(39,41)(44,46)(47,48);(17,21,23,22)(18,26,27,38)(19,31,30,32)(20,35,36,37)(24,28,40,45)(25,42,33,43)(29,47,34,48)(39,46,41,44)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6371132);