#501846

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

706 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)(7,9)(8,10)(11,12)(13,14,18,24)(15,25,19,34)(16,28,30,21)(17,22,33,36)(20,32,23,27)(26,40,31,39)(29,42,44,37)(35,41,38,43);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15)(14,20)(16,26)(18,19)(21,35)(22,25)(23,24)(27,32)(28,38)(30,31)(34,36)(37,39)(40,42)(41,43);(2,4)(6,10)(9,12)(13,16,18,30)(14,21,24,28)(15,26,19,31)(17,29,33,44)(20,35,23,38)(22,37,36,42)(25,39,34,40)(27,41,32,43);(14,22)(15,19)(17,32)(20,34)(21,37)(23,25)(24,36)(26,31)(27,33)(28,42)(29,43)(35,40)(38,39)(41,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,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^b=di,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409042);