#6459038

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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

1661 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)(21,34)(22,27)(23,30)(24,32)(26,36)(29,44)(31,42)(33,40)(35,39)(37,48)(38,46)(41,47)(43,45);(9,11)(10,13)(12,15)(14,16)(17,19,22,31)(18,23,27,39)(20,29,32,45)(21,35,36,30)(24,37,25,38)(26,42,34,28)(33,48,47,46)(40,44,41,43);(2,5)(6,8)(11,15)(13,16)(17,20,22,32)(18,24,27,25)(19,29,31,45)(21,33,36,47)(23,37,39,38)(26,40,34,41)(28,43,42,44)(30,46,35,48);(11,15)(13,16)(17,20)(18,25)(19,29)(21,33)(22,32)(23,38)(24,27)(26,41)(28,44)(30,46)(31,45)(34,40)(35,48)(36,47)(37,39)(42,43);(1,3,4,7)(2,6,5,8)(20,32)(24,25)(29,45)(33,47)(37,38)(40,41)(43,44)(46,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,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, 6366234);