#6459125

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1557 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)(9,10,12,14)(11,13,15,16)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,29)(30,43)(31,37)(32,39)(33,40)(34,41)(35,42)(38,44)(45,48)(46,47);(9,11)(10,13)(12,15)(14,16)(17,19,22,29,23,31,20,32)(18,24,25,28,27,39,36,37)(21,30,34,44,35,46,33,45)(26,38,40,43,42,48,41,47);(2,5)(6,8)(11,15)(13,16)(17,20,23,22)(18,25,27,36)(19,29,31,32)(21,34,35,33)(24,37,39,28)(26,41,42,40)(30,45,46,44)(38,43,48,47);(11,15)(13,16)(17,21)(18,26)(19,30)(20,33)(22,34)(23,35)(24,38)(25,40)(27,42)(28,43)(29,44)(31,46)(32,45)(36,41)(37,47)(39,48);(1,3,4,7)(2,6,5,8)(21,35)(26,42)(30,46)(33,34)(38,48)(40,41)(43,47)(44,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,d^c=di,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, 6366321);