#257964

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

771 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)(11,19,12,15)(14,23,22,18)(16,20,24,21)(25,26,28,31)(27,33,35,30)(29,38,37,32)(34,39,36,40);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,13)(14,20)(16,23)(17,19)(18,24)(21,22)(25,27)(26,30)(28,35)(29,34)(31,33)(32,39)(36,37)(38,40);(1,4,5,3)(2,6,7,8)(9,12,13,11)(10,16,17,24)(14,20,22,21)(15,18,19,23)(27,34)(29,37)(30,39)(32,38)(33,40)(35,36);(2,7)(6,8)(10,17)(11,12)(18,23)(20,21)(26,31)(30,33)(32,38)(39,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=ehi,c^2=ei,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, 165160);