#509809

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24266	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

711 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)(7,11)(8,10)(9,12)(13,14,18,24,15,25,27,39,17,23,31,38,19,20,33,22)(16,28,30,35,34,42,44,36,29,43,41,37,26,40,32,21);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,19)(14,20,23,25)(16,26,29,34)(18,27,31,33)(21,35,36,37)(22,38,39,24)(28,42,43,40)(30,32,41,44);(2,4)(13,16,17,29)(14,21,23,36)(15,26,19,34)(18,32,31,44)(20,35,25,37)(22,28,39,43)(24,40,38,42)(27,41,33,30);(6,10)(7,12)(14,22)(15,19)(18,33)(20,24)(21,28)(23,39)(25,38)(26,34)(27,31)(30,32)(35,40)(36,43)(37,42)(41,44)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417005);