#257969

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ehi,c^2=e,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, 165165);