#9851255

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

4415 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)(3,4)(5,7)(6,8)(9,10)(11,18)(12,20)(13,23)(14,16)(15,21)(17,22)(19,24)(25,26,31,34)(27,32,38,29)(28,30,40,33)(35,39,36,37);(1,3,6,7)(2,5,8,4)(9,11)(10,15)(12,19)(13,20)(14,21)(16,18)(17,24)(22,23)(25,27,31,38)(26,32,34,29)(28,36,40,35)(30,37,33,39);(10,16)(12,22)(15,18)(19,23)(25,28,31,40)(26,33,34,30)(27,35,38,36)(29,39,32,37);(1,4,6,5)(2,7,8,3)(9,12,14,22)(10,17,16,20)(11,19,21,23)(13,15,24,18)(25,28)(26,30)(27,36)(29,39)(31,40)(32,37)(33,34)(35,38);(1,5,6,4)(2,7,8,3)(9,11)(10,18)(12,19)(13,17)(14,21)(15,16)(20,24)(22,23)(25,29)(26,27)(28,39)(30,36)(31,32)(33,35)(34,38)(37,40);(1,2)(3,5)(4,7)(6,8)(9,13)(10,19)(11,20)(12,15)(14,24)(16,23)(17,21)(18,22)(25,30)(26,28)(27,37)(29,35)(31,33)(32,36)(34,40)(38,39)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=g,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, 9758451);