#6459014

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437428
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

1655 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)(19,27)(20,25)(21,33)(22,24)(23,29)(26,35)(28,41)(30,40)(31,38)(32,45)(34,36)(37,43)(39,44)(42,48)(46,47);(9,11)(10,13)(12,15)(14,16)(17,19,22,30)(18,23,24,36)(20,28,31,42)(21,34,35,29)(25,37,38,47)(26,40,33,27)(32,46,44,43)(39,48,45,41);(2,5)(6,8)(11,15)(13,16)(18,24)(20,31)(23,36)(26,33)(27,40)(28,42)(32,44)(43,46);(11,15)(13,16)(17,20,22,31)(18,25,24,38)(19,28,30,42)(21,32,35,44)(23,37,36,47)(26,39,33,45)(27,41,40,48)(29,43,34,46);(1,3,4,7)(2,6,5,8)(20,31)(25,38)(28,42)(32,44)(37,47)(39,45)(41,48)(43,46)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^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^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, 6366210);