#316840

order	512
nilpotency class	3
isoclinism	isoclinic class of #262851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

608 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,19)(14,18)(15,21)(16,22)(17,23)(20,24)(25,26)(27,35)(28,33)(29,32)(30,34)(31,37)(36,39)(38,40);(2,5)(6,7)(9,11,13,10)(12,15,20,17)(14,18,22,16)(19,23,24,21)(25,27,30,38,28,37,29,36)(26,31,34,39,33,35,32,40);(9,12,13,20)(10,15,11,17)(14,23,22,21)(16,24,18,19)(26,32)(27,36)(29,30)(31,35)(33,34)(37,38);(1,3,4,8)(2,6,5,7)(10,11)(15,17)(16,18)(19,24)(26,33)(27,37)(32,34)(36,38)

Polycyclic presentation:

2^9|b^2=g,c^2=hi,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^c=fh,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, 224036);