#9847972

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

4356 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=g,e^2=i,f^2=g,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, 9755168);