#512169

order	512
nilpotency class	4
isoclinism	isoclinic class of #511741
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

633 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=e,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,g^d=gh,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419365);