#9912914

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

4274 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)(3,4)(5,6)(7,8)(9,10,15,17)(11,16,20,24)(12,14,23,18)(13,19,21,22)(25,26,29,32)(27,30,33,37)(28,31,36,40)(34,38,35,39);(2,6)(3,7)(9,11,15,20)(10,16,17,24)(12,21,23,13)(14,22,18,19)(27,33)(30,37)(34,35)(38,39);(1,3,5,7)(2,4,6,8)(10,17)(11,20)(12,23)(19,22)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,36)(39,40);(1,4)(2,7)(3,6)(5,8)(9,12)(10,18)(11,21)(13,20)(14,17)(15,23)(16,19)(22,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,39)(36,40);(9,13,15,21)(10,19,17,22)(11,12,20,23)(14,24,18,16)(25,27)(26,30)(28,35)(29,33)(31,39)(32,37)(34,36)(38,40);(9,14,15,18)(10,12,17,23)(11,22,20,19)(13,16,21,24)(25,28)(26,31)(27,34)(29,36)(30,38)(32,40)(33,35)(37,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820110);