#501793

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×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

705 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)(7,9)(8,10)(11,12)(15,24)(20,35)(25,40)(26,28)(27,41)(32,43)(36,37)(39,44);(2,4)(5,7,8,11)(6,9,10,12)(13,14)(15,25)(16,21)(17,22)(18,31)(19,34)(20,27)(23,33)(24,40)(26,39)(28,44)(29,38)(30,42)(32,37)(35,41)(36,43);(6,10)(9,12)(13,15,17,24)(14,20,22,35)(16,28,19,26)(18,32,30,43)(21,37,34,36)(23,39,38,44)(25,42,40,31)(27,29,41,33);(2,4)(13,16,17,19)(14,21,22,34)(15,26,24,28)(18,29,30,33)(20,36,35,37)(23,31,38,42)(25,39,40,44)(27,43,41,32)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,h^c=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, 408989);