#6454076

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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×#3524	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3128 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)(7,10)(8,9)(11,12)(14,20)(15,25)(17,19)(21,34)(22,30)(23,24)(26,36)(27,39)(31,33)(35,43)(37,38)(40,44);(2,4)(13,14,18,23)(15,26,25,34)(16,30,32,37)(17,24,19,20)(21,28,36,29)(22,31,38,33)(27,35,39,44)(40,41,43,42);(6,9)(10,12)(13,15,18,25)(14,21,23,36)(16,27,32,39)(17,29,19,28)(20,26,24,34)(22,35,38,44)(30,40,37,43)(31,42,33,41);(13,16,18,32)(14,22,23,38)(15,27,25,39)(17,33,19,31)(20,30,24,37)(21,35,36,44)(26,40,34,43)(28,42,29,41);(5,7,8,11)(6,10,9,12)(15,25)(21,36)(26,34)(27,39)(28,29)(35,44)(40,43)(41,42)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=i,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,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, 6361272);