#9065342

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
prime radical	#35932	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4411 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8972538);