#9912909

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

4251 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^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, 9820105);