#500871

order	512
nilpotency class	4
isoclinism	isoclinic class of #499891
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

802 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,f^c=fh,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, 408067);