#315312

order	512
nilpotency class	3
isoclinism	isoclinic class of #288787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

520 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)(11,16)(12,21)(13,23)(14,18)(15,19)(17,22)(20,24)(25,26,30,35,31,36,29,34)(27,32,37,40,28,39,38,33);(1,3,4,7)(2,5,6,8)(9,11,13,19,14,20,22,12)(10,15,17,16,18,21,23,24)(25,27,29,38,31,28,30,37)(26,32,34,33,36,39,35,40);(2,6)(5,8)(25,28,31,27)(26,33,36,40)(29,38,30,37)(32,35,39,34);(9,12,22,20,14,19,13,11)(10,16,23,15,18,24,17,21)(25,29,31,30)(26,34,36,35)(27,37,28,38)(32,40,39,33)

Polycyclic presentation:

2^9|a^2=g,b^2=ef,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,f^c=fi,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, 222508);