#9847960

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

4444 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=g,e^2=gi,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, 9755156);