#9243396

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

4417 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=g,d^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, 9150592);