#6454022

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×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3088 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)(13,14,19,23)(15,26,29,39)(16,32,17,22)(18,25,20,24)(21,31,38,30)(27,43,28,40)(33,44,36,37)(34,41,35,42);(2,4)(13,15,19,29)(14,21,23,38)(16,33,17,36)(18,31,20,30)(22,40,32,43)(24,39,25,26)(27,34,28,35)(37,41,44,42);(6,9)(10,12)(13,16,19,17)(14,22,23,32)(15,27,29,28)(18,35,20,34)(21,37,38,44)(24,42,25,41)(26,40,39,43)(30,33,31,36);(13,17,19,16)(14,22,23,32)(15,28,29,27)(18,34,20,35)(21,37,38,44)(24,42,25,41)(26,40,39,43)(30,36,31,33);(5,7,8,11)(6,10,9,12)(14,23)(15,29)(22,32)(24,25)(27,28)(30,31)(33,36)(41,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^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=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, 6361218);