#9674980

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

4411 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=gh,e^2=g,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, 9582176);