#9161739

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

4552 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=h,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9068935);