#6439446

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2328 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi

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, 6346642);