#280839

order	512
nilpotency class	3
isoclinism	isoclinic class of #244927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fhi,c^2=hi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,g^b=gi,d^c=di,g^c=gi,e^d=eh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 188035);