#9815218

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

4497 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=hi,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, 9722414);