#6459088

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1571 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)(9,10,12,14)(11,13,15,16)(17,18)(19,28)(20,25)(21,26)(22,37)(23,27)(24,30)(29,44)(31,39)(32,46)(33,42)(34,43)(35,48)(36,40)(38,45)(41,47);(9,11)(10,13)(12,15)(14,16)(17,19,22,30,23,31,21,36)(18,24,26,28,27,40,37,39)(20,32,35,47,34,29,33,45)(25,41,42,46,43,38,48,44);(2,5)(6,8)(11,15)(13,16)(17,20)(18,25)(19,29)(21,33)(22,35)(23,34)(24,38)(26,42)(27,43)(28,44)(30,45)(31,32)(36,47)(37,48)(39,46)(40,41);(11,15)(13,16)(17,21,23,22)(18,26,27,37)(19,30,31,36)(20,33,34,35)(24,39,40,28)(25,42,43,48)(29,45,32,47)(38,46,41,44);(1,3,4,7)(2,6,5,8)(20,34)(25,43)(29,32)(33,35)(38,41)(42,48)(44,46)(45,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,d^2=i,e^2=g,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, 6366284);