#9675017

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

4349 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)(11,18)(12,20)(13,15)(14,17)(16,22)(19,23)(21,24)(26,32)(27,30)(28,34)(37,39);(1,2,5,8)(3,4,7,6)(9,11,14,13)(10,15,17,18)(12,19,22,21)(16,23,20,24)(25,26)(27,35)(28,29)(30,33)(31,32)(34,40)(36,37)(38,39);(1,3)(2,6)(4,8)(5,7)(25,27,31,30)(26,33,32,35)(28,38,34,36)(29,37,40,39);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)(25,28,31,34)(26,29,32,40)(27,36,30,38)(33,39,35,37);(1,3)(2,4)(5,7)(6,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,23)(20,22)(21,24)(25,29)(26,34)(27,37)(28,32)(30,39)(31,40)(33,36)(35,38);(1,4)(2,7)(3,8)(5,6)(9,13,14,11)(10,15,17,18)(12,21,22,19)(16,23,20,24)(25,30,31,27)(26,33,32,35)(28,38,34,36)(29,39,40,37)

Polycyclic presentation:

2^9|b^2=gi,c^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,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, 9582213);