#9923992

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

4266 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=ghi,d^2=gi,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, 9831188);