#9936406

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

4494 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)(5,8)(6,7)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26,31,36)(27,34,39,32)(28,33,38,29)(30,35,40,37);(1,3)(2,4)(5,6)(7,8)(12,19)(15,17)(20,24)(22,23)(25,27,31,39)(26,32,36,34)(28,30,38,40)(29,35,33,37);(1,4)(2,6)(3,8)(5,7)(10,11)(15,17)(16,18)(22,23)(25,28,31,38)(26,33,36,29)(27,30,39,40)(32,37,34,35);(1,2,5,8)(3,7,6,4)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,29)(26,28)(27,37)(30,34)(31,33)(32,40)(35,39)(36,38);(1,4)(2,3)(5,7)(6,8)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,27)(26,34)(28,30)(29,37)(31,39)(32,36)(33,35)(38,40);(1,3)(2,7)(4,8)(5,6)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30)(26,35)(27,38)(28,39)(29,34)(31,40)(32,33)(36,37)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,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, 9843602);