#9717555

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

4502 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)(3,4)(5,6)(7,8)(9,10)(11,18)(12,16)(13,21)(14,15)(17,23)(19,20)(22,24)(28,36)(31,38)(35,40)(37,39);(2,6)(3,7)(9,10)(11,12)(13,17)(14,15)(16,18)(19,20)(21,23)(22,24)(25,26)(27,32)(28,31)(29,33)(30,34)(35,39)(36,38)(37,40);(1,3,5,7)(2,4,6,8)(10,15)(11,16)(17,21)(19,24)(25,27)(26,29)(28,35)(30,33)(31,37)(32,34)(36,40)(38,39);(1,4)(2,7)(3,6)(5,8)(9,11,14,16)(10,12,15,18)(13,19,23,24)(17,20,21,22)(25,28)(26,31)(27,35)(29,37)(30,36)(32,39)(33,40)(34,38);(9,12)(10,16)(11,15)(13,22)(14,18)(17,19)(20,23)(21,24)(25,27)(26,32)(28,35)(29,34)(30,33)(31,39)(36,40)(37,38);(9,13)(10,17)(11,19)(12,20)(14,23)(15,21)(16,24)(18,22)(25,29,30,32)(26,33,34,27)(28,37,36,39)(31,40,38,35)

Polycyclic presentation:

2^9|c^2=g,d^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9624751);