#9183875

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

4457 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=hi,d^2=gh,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=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, 9091071);