#10360958

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4645 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=h,d^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10268154);