#9203785

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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,e^2=gi,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, 9110981);