#9909756

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

4621 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,d^2=gi,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9816952);