#501842

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

710 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)(13,14,19,25)(15,24,18,33)(16,21,30,39)(17,22,32,35)(20,31,23,27)(26,38,29,43)(28,36,42,44)(34,41,37,40);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15)(14,20)(16,26)(18,19)(21,34)(22,24)(23,25)(27,31)(29,30)(33,35)(36,38)(37,39)(40,41)(43,44);(2,4)(6,10)(9,12)(13,16,19,30)(14,21,25,39)(15,26,18,29)(17,28,32,42)(20,34,23,37)(22,36,35,44)(24,38,33,43)(27,40,31,41);(14,22)(15,18)(17,31)(20,33)(21,36)(23,24)(25,35)(26,29)(27,32)(28,41)(34,43)(37,38)(39,44)(40,42)

Polycyclic presentation:

2^9|a^2=i,b^2=f,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, 409038);