#9682875

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

4170 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=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, 9590071);