#6468684

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1547 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)(10,14)(15,16)(17,18,24,40,23,29,22,28)(19,30,39,25,34,48,33,43)(20,26,21,27,36,45,37,46)(31,42,38,41,35,47,32,44);(9,10,13,14)(11,16,12,15)(17,19,23,34)(18,25,29,43)(20,35,36,31)(21,38,37,32)(22,39,24,33)(26,44,45,41)(27,47,46,42)(28,48,40,30);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,23,36)(18,26,29,45)(19,31,34,35)(21,22,37,24)(25,41,43,44)(27,28,46,40)(30,42,48,47)(32,39,38,33);(9,12,13,11)(10,16,14,15)(17,21)(18,27)(19,32)(20,22)(23,37)(24,36)(25,42)(26,28)(29,46)(30,44)(31,39)(33,35)(34,38)(40,45)(41,48)(43,47);(1,3,4,7)(2,6,5,8)(20,36)(21,37)(26,45)(27,46)(31,35)(32,38)(41,44)(42,47)

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=h,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=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, 6375880);