#9913215

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

4767 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820411);