#500321

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

774 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=e,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,f^c=fh,g^c=gh,h^c=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, 407517);