#9923636

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

4217 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)(4,6,8,7)(9,10,14,15)(11,13,17,22)(12,16,20,23)(18,21,24,19)(25,26)(27,31)(28,37)(29,33)(30,34)(32,36)(35,40)(38,39);(2,3)(6,7)(9,11)(10,13)(12,18)(14,17)(15,22)(16,21)(19,23)(20,24)(27,35)(29,38)(31,40)(33,39);(10,15)(11,17)(16,23)(18,24)(25,27,30,35)(26,31,34,40)(28,29,36,38)(32,39,37,33);(9,10)(11,13)(12,16)(14,15)(17,22)(18,21)(19,24)(20,23)(25,28,30,36)(26,32,34,37)(27,29,35,38)(31,39,40,33);(1,3,5,2)(4,6,8,7)(9,12,14,20)(10,16,15,23)(11,18,17,24)(13,21,22,19)(25,29)(26,33)(27,36)(28,35)(30,38)(31,32)(34,39)(37,40);(1,4,5,8)(2,6,3,7)(9,13)(10,11)(12,19)(14,22)(15,17)(16,24)(18,23)(20,21)(25,26)(27,31)(28,32)(29,39)(30,34)(33,38)(35,40)(36,37)

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=hi,e^2=gh,f^2=g,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, 9830832);