#311319

order	512
nilpotency class	3
isoclinism	isoclinic class of #264203
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

557 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)(3,8)(4,6)(5,7)(9,10,15,20)(11,17,22,18)(12,23,13,16)(14,24,21,19)(25,26)(27,34)(28,38)(29,33)(30,32)(31,37)(35,40)(36,39);(1,3,5,4)(2,6,7,8)(9,11,21,12,15,22,14,13)(10,16,24,18,20,23,19,17)(25,27)(26,31)(28,36)(29,37)(30,35)(32,39)(33,34)(38,40);(1,4,5,3)(2,6,7,8)(9,12,14,11,15,13,21,22)(10,17,19,23,20,18,24,16)(27,35)(28,30)(31,39)(32,38)(34,40)(36,37);(2,7)(6,8)(9,13,14,22,15,12,21,11)(10,18,19,16,20,17,24,23)(25,28,29,30)(26,32,33,38)(27,36,37,35)(31,39,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 218515);