#9229740

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

4556 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9136936);