#9815358

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

4166 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=ghi,f^2=h,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, 9722554);