#9947514

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

4503 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=ghi,e^2=gh,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, 9854710);