#9170462

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

4772 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)(9,10,14,18)(11,15,19,16)(12,13,17,23)(20,21,22,24)(25,26,30,28)(27,32,35,31)(29,36,37,33)(34,38,40,39);(3,7)(6,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,18)(17,22)(23,24)(26,28)(31,32)(33,36)(38,39);(1,2)(3,6)(4,5)(7,8)(11,19)(15,16)(20,22)(21,24)(25,27,30,35)(26,31,28,32)(29,34,37,40)(33,38,36,39);(9,11)(10,16)(12,22)(13,21)(14,19)(15,18)(17,20)(23,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,13)(11,20)(14,17)(15,21)(16,24)(18,23)(19,22)(25,28,30,26)(27,32,35,31)(29,33,37,36)(34,38,40,39);(1,2)(3,6)(4,5)(7,8)(9,13)(10,17)(11,21)(12,18)(14,23)(15,22)(16,20)(19,24)(25,29,30,37)(26,33,28,36)(27,34,35,40)(31,38,32,39)

Polycyclic presentation:

2^9|a^2=gi,c^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9077658);