#9160912

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	#40157	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4315 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)(11,15)(12,16)(13,24)(14,18)(17,21)(19,20)(22,23)(25,26,30,33)(27,34,29,31)(28,36,38,32)(35,39,37,40);(1,2)(3,8)(4,6)(5,7)(9,11,14,20)(10,15,18,19)(12,21,23,13)(16,17,22,24)(25,26)(27,31)(28,32)(29,34)(30,33)(35,39)(36,38)(37,40);(1,3)(2,4)(5,6)(7,8)(9,10,14,18)(11,19,20,15)(12,16,23,22)(13,17,21,24)(25,27,30,29)(26,31,33,34)(28,37,38,35)(32,40,36,39);(3,6)(4,8)(9,10)(11,15)(12,22)(13,17)(14,18)(16,23)(19,20)(21,24)(25,28,30,38)(26,32,33,36)(27,35,29,37)(31,39,34,40);(1,4,5,8)(2,6,7,3)(9,12,14,23)(10,16,18,22)(11,13,20,21)(15,24,19,17)(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)(10,17)(11,12)(14,21)(15,22)(16,19)(18,24)(20,23)(25,26)(27,31)(28,32)(29,34)(30,33)(35,39)(36,38)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=g,e^2=gi,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, 9068108);