#499459

order	512
nilpotency class	4
isoclinism	isoclinic class of #498795
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×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

632 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)(7,10)(8,9)(11,12)(13,14)(15,25)(16,22)(17,21)(18,23)(19,24)(20,28)(26,34)(27,41)(29,42)(30,38)(31,37)(32,40)(33,35)(36,43)(39,44);(6,9)(10,12)(13,15,17,28)(14,20,21,25)(16,27,19,33)(18,26,31,44)(22,35,24,41)(23,39,37,34)(29,30,43,32)(36,38,42,40);(2,4)(14,21)(15,26)(16,19)(18,32)(20,34)(23,38)(25,39)(27,29)(28,44)(30,31)(33,43)(35,42)(36,41)(37,40);(5,7,8,11)(6,10,9,12)(13,16,17,19)(14,22,21,24)(15,27,28,33)(18,30,31,32)(20,35,25,41)(23,38,37,40)(26,43,44,29)(34,36,39,42)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,h^c=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, 406655);