#280170

order	512
nilpotency class	3
isoclinism	isoclinic class of #249463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

611 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 187366);