#9794222

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

4492 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gh,d^2=h,e^2=h,f^2=gh,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, 9701418);