#9726984

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	#47951	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4444 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9634180);