#6445473

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2996 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)(5,6,8,10)(7,11,12,9)(13,14)(15,25)(16,21)(17,24)(18,23)(19,22)(20,26)(27,41)(28,35)(29,36)(30,44)(31,40)(32,39)(33,38)(34,42)(37,43);(2,4)(13,15,19,29)(14,20,22,35)(16,30,33,42)(17,28,18,26)(21,37,38,41)(23,36,24,25)(27,32,43,31)(34,40,44,39);(5,7,8,12)(6,9,10,11)(15,26)(17,18)(20,25)(23,24)(27,42)(28,29)(30,43)(31,32)(34,41)(35,36)(37,44)(39,40);(13,16,19,33)(14,21,22,38)(15,27,29,43)(17,32,18,31)(20,34,35,44)(23,40,24,39)(25,41,36,37)(26,42,28,30);(14,22)(20,35)(21,38)(23,24)(25,36)(34,44)(37,41)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^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=dhi,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352669);