#508220

order	512
nilpotency class	4
isoclinism	isoclinic class of #507998
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

778 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:

(6,7)(10,11)(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)(5,6)(7,9)(8,10)(11,12)(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)(5,7,9,6)(8,10,12,11)(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);(5,8,9,12)(6,10,7,11)(16,28)(21,38)(25,34)(27,43)(29,37)(32,44)(35,41)(40,42)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415416);