#9794246

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

4435 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)(9,10,14,16)(11,18,22,15)(12,21,23,17)(13,20,24,19)(25,26)(27,35)(28,37)(29,34)(30,38)(31,33)(32,36)(39,40);(1,2,6,4)(3,8,5,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,18)(17,24)(19,23)(25,27,31,36)(26,32,33,35)(28,29,38,39)(30,34,37,40);(1,3,6,5)(2,7,4,8)(10,16)(12,23)(15,18)(19,20)(26,33)(27,36)(28,38)(34,40);(1,2,6,4)(3,8,5,7)(9,12)(10,17)(11,20)(13,18)(14,23)(15,24)(16,21)(19,22)(25,28)(26,30)(27,29)(31,38)(32,34)(33,37)(35,40)(36,39);(1,4,6,2)(3,8,5,7)(9,11)(10,18)(12,20)(13,17)(14,22)(15,16)(19,23)(21,24)(25,29)(26,34)(27,28)(30,32)(31,39)(33,40)(35,37)(36,38);(1,5,6,3)(2,7,4,8)(9,13)(10,19)(11,21)(12,18)(14,24)(15,23)(16,20)(17,22)(25,30)(26,28)(27,34)(29,32)(31,37)(33,38)(35,39)(36,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gh,d^2=gh,e^2=gh,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, 9701442);