#501830

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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

713 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)(13,14,17,22)(15,24,26,20)(16,19,30,33)(18,31,29,23)(21,37,34,28)(25,38,43,32)(27,36,42,40)(35,41,39,44);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(14,19)(16,28)(18,29)(20,32)(21,23)(22,33)(24,38)(25,41)(27,42)(30,37)(31,34)(35,36)(39,40)(43,44);(2,4)(6,10)(9,12)(13,15,17,26)(14,20,22,24)(16,25,30,43)(18,27,29,42)(19,32,33,38)(21,35,34,39)(23,36,31,40)(28,41,37,44);(14,19)(16,28)(18,29)(20,32)(21,23)(22,33)(24,38)(25,41)(27,42)(30,37)(31,34)(35,36)(39,40)(43,44)

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,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409026);