#9057798

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

4490 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)(3,7,8,4)(9,10)(11,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,24)(25,26,30,34)(27,32,31,28)(29,37,39,33)(35,36,40,38);(2,6)(4,7)(10,15)(12,19)(16,22)(20,24)(25,27)(26,28)(29,35)(30,31)(32,34)(33,36)(37,38)(39,40);(1,2)(3,4)(5,6)(7,8)(11,17)(12,19)(13,21)(16,22)(25,28,30,32)(26,31,34,27)(29,38,39,36)(33,35,37,40);(1,3)(2,4)(5,8)(6,7)(9,11,14,17)(10,12,15,19)(13,18,21,23)(16,20,22,24)(25,28)(26,27)(29,36)(30,32)(31,34)(33,35)(37,40)(38,39);(1,4)(2,3)(5,7)(6,8)(9,12)(10,11)(13,20)(14,19)(15,17)(16,18)(21,24)(22,23)(25,27)(26,32)(28,34)(29,35)(30,31)(33,38)(36,37)(39,40);(1,3)(2,4)(5,8)(6,7)(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,20,19,24)(25,29,30,39)(26,33,34,37)(27,35,31,40)(28,36,32,38)

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=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, 8964994);