#9189477

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	#51040	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4472 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9096673);