#6448836

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3274 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,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, 6356032);