#9851251

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

4540 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,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, 9758447);