#6454038

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×#3524	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3136 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)(13,14)(15,25)(16,22)(17,24)(18,21)(19,23)(20,26)(27,41)(28,34)(29,36)(30,44)(31,40)(32,37)(33,39)(35,42)(38,43);(2,4)(13,15,18,28)(14,20,21,36)(16,30,32,42)(17,29,19,26)(22,38,37,41)(23,34,24,25)(27,33,43,31)(35,40,44,39);(6,9)(10,12)(14,21)(15,26)(17,19)(20,34)(22,37)(25,36)(27,42)(28,29)(30,43)(31,33)(35,38)(41,44);(13,16,18,32)(14,22,21,37)(15,27,28,43)(17,33,19,31)(20,35,36,44)(23,40,24,39)(25,41,34,38)(26,42,29,30);(5,7,8,11)(6,10,9,12)(14,21)(15,28)(22,37)(23,24)(26,29)(27,43)(30,42)(39,40)

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=gi,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,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, 6361234);