#9203735

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

4513 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9110931);