#505693

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

539 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)(5,6,8,10)(7,9,11,12)(13,14,18,23)(15,25,27,43)(16,30,29,44)(17,33,35,42)(19,36,34,24)(20,37,38,28)(21,40,41,32)(22,31,39,26);(5,7)(6,9)(8,11)(10,12)(13,15,26,40,19,29,35,37,18,27,31,32,34,16,17,28)(14,20,33,30,24,21,39,43,23,38,42,44,36,41,22,25);(2,4)(7,11)(9,12)(15,27)(16,29)(20,38)(21,41)(25,43)(28,37)(30,44)(32,40);(13,16,31,37,19,15,17,32,18,29,26,28,34,27,35,40)(14,21,42,25,24,38,22,30,23,41,33,43,36,20,39,44)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=ehi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^c=di

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, 412889);