#317457

order	512
nilpotency class	3
isoclinism	isoclinic class of #262915
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

578 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)(3,7)(4,5)(6,8)(9,10)(11,17)(12,15)(13,16)(14,20)(18,22)(19,24)(21,23)(25,26,28,32)(27,35,37,30)(29,34,36,33)(31,39,40,38);(2,5)(6,7)(9,11,18,19,12,20,13,21)(10,14,22,23,15,17,16,24)(25,27,36,39,28,37,29,38)(26,30,33,31,32,35,34,40);(11,19)(13,18)(14,23)(16,22)(17,24)(20,21)(25,27,28,37)(26,31,32,40)(29,39,36,38)(30,34,35,33);(1,3,4,8)(2,6,5,7)(10,15)(11,20)(16,22)(19,21)(26,32)(30,35)(31,40)(33,34)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dhi,f^c=fh,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, 224653);