#9717756

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

4645 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=h,e^2=h,f^2=h,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, 9624952);