#7658255

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

4782 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565451);