#280841

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

637 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)(25,26,29,30)(27,31,34,38)(28,35,37,32)(33,36,40,39);(1,3,4,7)(2,5,6,8)(10,15)(12,13)(16,23)(17,18)(19,24)(20,21)(25,27)(26,31)(28,36)(29,34)(30,38)(32,33)(35,40)(37,39);(2,6)(5,8)(9,11,14,22)(10,16,18,19)(12,20,13,21)(15,23,17,24)(25,28,29,37)(26,32,30,35)(27,33,34,40)(31,39,38,36);(9,12)(11,20)(13,14)(15,17)(21,22)(23,24)(25,26,29,30)(27,31,34,38)(28,35,37,32)(33,36,40,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=fhi,c^2=hi,d^2=i,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, 188037);