#9947164

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

4577 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)(3,5)(4,6)(7,8)(9,10,14,17)(11,15,20,23)(12,13,21,16)(18,19,24,22)(27,33)(30,34)(35,39)(38,40);(1,2,4,6)(3,5,8,7)(9,11,14,20)(10,15,17,23)(12,18,21,24)(13,19,16,22)(25,26,29,32)(27,34,33,30)(28,31,37,36)(35,40,39,38);(9,12,14,21)(10,16,17,13)(11,18,20,24)(15,22,23,19)(25,26)(27,30)(28,31)(29,32)(33,34)(35,38)(36,37)(39,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,13)(11,18)(14,21)(15,19)(16,17)(20,24)(22,23)(25,27,29,33)(26,30,32,34)(28,35,37,39)(31,38,36,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24)(25,26)(27,30)(28,36)(29,32)(31,37)(33,34)(35,40)(38,39);(1,3,4,8)(2,5,6,7)(9,13)(10,12)(11,19)(14,16)(15,18)(17,21)(20,22)(23,24)(25,28)(26,31)(27,35)(29,37)(30,38)(32,36)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=i,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, 9854360);