#506189

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	#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:

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

Polycyclic presentation:

2^9|b^2=eg,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,g^c=gh,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, 413385);