#6469057

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1848 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)(3,8)(4,6)(5,7)(11,15)(13,16)(17,18)(19,28)(20,25)(21,35)(22,27)(23,30)(24,33)(26,37)(29,45)(31,43)(32,44)(34,41)(36,40)(38,47)(39,46)(42,48);(9,10)(11,16)(12,14)(13,15)(17,19,22,31)(18,23,27,40)(20,32,33,29)(21,36,37,30)(24,39,25,38)(26,43,35,28)(34,46,48,47)(41,44,42,45);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,20)(18,24)(19,29)(21,34)(22,33)(23,38)(25,27)(26,41)(28,44)(30,46)(31,32)(35,42)(36,47)(37,48)(39,40)(43,45);(10,14)(13,16)(17,20)(18,25)(19,29)(21,34)(22,33)(23,39)(24,27)(26,42)(28,45)(30,46)(31,32)(35,41)(36,47)(37,48)(38,40)(43,44);(1,4,5,3)(2,6,7,8)(20,33)(24,25)(29,32)(34,48)(38,39)(41,42)(44,45)(46,47)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,e^2=gh,f^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6376253);