#9271799

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

4299 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)(11,17)(12,21)(13,15)(14,18)(16,19)(20,22)(23,24)(25,26,32,36)(27,30,28,33)(29,38,39,34)(31,37,40,35);(1,2)(3,4)(5,6)(7,8)(9,11,16,13)(10,17,19,15)(12,20,14,23)(18,24,21,22)(25,27,32,28)(26,33,36,30)(29,37,39,35)(31,34,40,38);(2,6)(3,7)(9,12,16,14)(10,18,19,21)(11,20,13,23)(15,22,17,24)(25,28,32,27)(26,30,36,33)(29,37,39,35)(31,34,40,38);(9,13,16,11)(10,15,19,17)(12,20,14,23)(18,24,21,22)(25,29)(26,34)(27,37)(28,35)(30,31)(32,39)(33,40)(36,38);(1,3,5,7)(2,4,6,8)(9,14,16,12)(10,18,19,21)(11,20,13,23)(15,24,17,22)(25,30)(26,28)(27,36)(29,31)(32,33)(34,35)(37,38)(39,40);(1,4)(2,7)(3,6)(5,8)(9,15)(10,11)(12,22)(13,19)(14,24)(16,17)(18,20)(21,23)(25,31)(26,35)(27,38)(28,34)(29,30)(32,40)(33,39)(36,37)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=gh,e^2=ghi,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, 9178995);