#257970

order	512
nilpotency class	3
isoclinism	isoclinic class of #247367
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

753 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)(3,8)(4,6)(5,7)(9,10,13,17)(11,19,21,15)(12,18,14,16)(20,24,22,23)(27,35)(28,29)(31,39)(33,34)(36,40)(37,38);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,22)(13,21)(14,20)(16,23)(17,19)(18,24)(25,26)(27,36)(28,34)(29,33)(30,32)(31,38)(35,40)(37,39);(1,4,5,3)(2,6,7,8)(10,16)(12,14)(15,23)(17,18)(19,24)(20,22)(25,27,30,37)(26,31,32,40)(28,38,29,35)(33,36,34,39);(2,7)(6,8)(9,12,13,14)(10,16,17,18)(11,20,21,22)(15,24,19,23)(26,32)(27,37)(33,34)(35,38)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 165166);