#9236432

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

4302 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9143628);