#8182066

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

4652 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:

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

Polycyclic presentation:

2^9|b^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8089262);