#256513

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

723 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)(9,10,12,17)(11,19,21,15)(13,16,14,18)(20,24,22,23)(25,26,31,35)(27,32,36,33)(28,37,38,40)(29,34,30,39);(1,3,5,4)(2,6,7,8)(9,11,12,21)(10,15,17,19)(13,22,14,20)(16,24,18,23)(25,27)(26,32)(28,29)(30,38)(31,36)(33,35)(34,37)(39,40);(1,4,5,3)(2,6,7,8)(10,16)(11,20)(13,14)(17,18)(21,22)(23,24)(25,28,31,38)(26,33,35,32)(27,29,36,30)(34,40,39,37);(2,7)(6,8)(10,17)(15,19)(16,18)(23,24)(25,29,31,30)(26,34,35,39)(27,28,36,38)(32,37,33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 163709);