#9232797

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

4546 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)(9,10,15,18)(11,12,13,16)(14,22,24,17)(19,23,21,20)(25,26,31,34)(27,32,37,29)(28,38,30,33)(35,39,36,40);(1,3,5,8)(2,6,7,4)(9,11,15,13)(10,16,18,12)(14,19,24,21)(17,23,22,20)(25,27,31,37)(26,32,34,29)(28,36,30,35)(33,39,38,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,18)(14,20)(15,16)(17,19)(21,22)(23,24)(25,28,31,30)(26,33,34,38)(27,35,37,36)(29,39,32,40);(1,4)(2,3)(5,6)(7,8)(9,13,15,11)(10,12,18,16)(14,19,24,21)(17,23,22,20)(25,29)(26,27)(28,39)(30,40)(31,32)(33,35)(34,37)(36,38);(1,3)(2,4)(5,8)(6,7)(9,12)(10,13)(11,18)(14,23)(15,16)(17,19)(20,24)(21,22)(25,30,31,28)(26,33,34,38)(27,36,37,35)(29,39,32,40);(1,4)(2,3)(5,6)(7,8)(9,14,15,24)(10,17,18,22)(11,19,13,21)(12,20,16,23)(25,26)(27,32)(28,33)(29,37)(30,38)(31,34)(35,40)(36,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=hi,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9139993);