#9946955

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

4685 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=g,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9854151);