#9267048

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

4752 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)(3,4)(5,6)(7,8)(10,11)(15,17)(16,18)(22,23)(25,26)(27,33)(28,32)(29,34)(30,31)(35,40)(36,37)(38,39);(2,6)(3,7)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,27)(26,29)(28,36)(30,34)(31,33)(32,40)(35,38)(37,39);(1,3,5,7)(2,4,6,8)(12,19)(15,17)(20,24)(22,23)(26,31)(27,34)(32,39)(35,36);(1,4)(2,7)(3,6)(5,8)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,38)(39,40);(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28,30,38)(26,32,31,39)(27,35,34,36)(29,37,33,40);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,27)(28,37)(30,33)(31,34)(32,35)(36,39)(38,40)

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9174244);