#9851163

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

4895 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,e^2=ghi,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, 9758359);