#506204

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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

719 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)(13,14)(15,20)(16,27)(17,22)(18,21)(19,23)(24,35)(25,34)(26,36)(28,33)(29,41)(30,38)(31,37)(32,39)(40,44)(42,43);(5,7,8,12)(6,9,10,11)(13,15,17,25,16,28,29,42,19,26,32,40,18,24,31,30)(14,20,22,34,27,33,41,43,23,36,39,44,21,35,37,38);(2,4)(13,15)(14,20)(16,24)(17,30)(18,28)(19,26)(21,33)(22,38)(23,36)(25,31)(27,35)(29,40)(32,42)(34,37)(39,43)(41,44);(6,10)(9,11)(13,16,19,18)(14,21,23,27)(15,24,26,28)(17,29,32,31)(20,33,36,35)(22,37,39,41)(25,30,40,42)(34,43,44,38)

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,d^b=di,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, 413400);