#9847901

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

4779 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)(12,19)(13,14)(16,21)(17,24)(18,22)(20,23)(25,26)(27,36)(28,30)(29,32)(31,33)(34,35)(37,40)(38,39);(2,3)(6,7)(9,10)(11,15)(12,16)(13,14)(17,23)(18,22)(19,21)(20,24)(25,27,31,29)(26,32,33,36)(28,37,35,39)(30,38,34,40);(1,2,5,3)(4,6,8,7)(10,14)(15,22)(16,19)(23,24)(26,33)(28,35)(32,36)(37,39);(9,10)(11,15)(12,16)(13,14)(17,23)(18,22)(19,21)(20,24)(25,28,31,35)(26,34,33,30)(27,37,29,39)(32,40,36,38);(1,3,5,2)(4,6,8,7)(9,11,13,18)(10,15,14,22)(12,20,21,17)(16,24,19,23)(25,29,31,27)(26,32,33,36)(28,39,35,37)(30,38,34,40);(1,4)(2,6)(3,7)(5,8)(9,12)(10,16)(11,17)(13,21)(14,19)(15,23)(18,20)(22,24)(25,30,31,34)(26,35,33,28)(27,38,29,40)(32,39,36,37)

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,d^2=g,e^2=gi,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, 9755097);