#8581445

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

4509 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)(4,8)(5,6)(9,10,15,12)(11,13,16,18)(14,17,22,20)(19,21,23,24)(25,26,31,30)(27,32,37,36)(28,38,39,33)(29,34,35,40);(2,6)(3,8)(9,11)(10,13)(12,18)(14,19)(15,16)(17,21)(20,24)(22,23)(25,27,31,37)(26,32,30,36)(28,29,39,35)(33,40,38,34);(1,3)(2,4)(5,8)(6,7)(9,10,15,12)(11,18,16,13)(14,17,22,20)(19,24,23,21)(25,28,31,39)(26,33,30,38)(27,35,37,29)(32,34,36,40);(1,3)(2,4)(5,8)(6,7)(9,12,15,10)(11,18,16,13)(14,17,22,20)(19,21,23,24)(25,29)(26,34)(27,28)(30,40)(31,35)(32,38)(33,36)(37,39);(1,4)(2,3)(5,7)(6,8)(9,13,15,18)(10,16,12,11)(14,21,22,24)(17,23,20,19)(25,28)(26,33)(27,29)(30,38)(31,39)(32,40)(34,36)(35,37);(9,14,15,22)(10,17,12,20)(11,19,16,23)(13,21,18,24)(25,30,31,26)(27,36,37,32)(28,38,39,33)(29,34,35,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=hi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

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, 8488641);