#311508

order	512
nilpotency class	3
isoclinism	isoclinic class of #264283
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

522 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)(9,10,15,21)(11,22,24,18)(12,16,23,17)(13,20,14,19)(25,26,30,34)(27,31,37,40)(28,32,29,33)(35,38,36,39);(1,3,5,4)(2,6,7,8)(9,11,14,23,15,24,13,12)(10,16,20,18,21,17,19,22)(25,27,30,37)(26,31,34,40)(28,36,29,35)(32,39,33,38);(1,4,5,3)(2,6,7,8)(9,11,13,12,15,24,14,23)(10,17,19,18,21,16,20,22)(27,35)(28,29)(31,38)(32,33)(36,37)(39,40);(2,7)(6,8)(9,12,13,24,15,23,14,11)(10,18,19,16,21,22,20,17)(25,28,30,29)(26,32,34,33)(27,36,37,35)(31,39,40,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=ehi,c^2=e,d^2=fhi,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=dhi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 218704);