#9947497

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

4525 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)(9,10,14,11)(12,15,20,17)(13,18,22,16)(19,24,21,23)(25,26)(27,29)(28,32)(30,31)(33,37)(34,36)(35,38)(39,40);(1,2,6,5)(3,4,8,7)(10,11)(15,17)(16,18)(23,24)(28,35)(32,38)(34,39)(36,40);(1,3)(2,7)(4,5)(6,8)(10,11)(15,17)(16,18)(23,24)(26,31)(27,33)(32,38)(34,39);(1,4)(2,3)(5,8)(6,7)(9,11,14,10)(12,17,20,15)(13,18,22,16)(19,24,21,23)(25,27,30,33)(26,29,31,37)(28,34,35,39)(32,36,38,40);(1,5,6,2)(3,4,8,7)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,28,30,35)(26,32,31,38)(27,34,33,39)(29,36,37,40);(1,4)(2,8)(3,5)(6,7)(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,29)(26,33)(27,31)(28,36)(30,37)(32,39)(34,38)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=hi,e^2=h,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, 9854693);