#9718547

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

4573 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)(3,6)(4,7)(5,8)(9,10)(11,12)(13,21)(14,15)(16,18)(17,23)(19,22)(20,24)(25,26,30,31)(27,33,35,28)(29,32,38,39)(34,40,37,36);(1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,19)(14,18)(15,16)(17,20)(21,22)(23,24)(25,27)(26,28)(29,37)(30,35)(31,33)(32,40)(34,38)(36,39);(10,15)(11,18)(17,21)(19,24)(26,31)(28,33)(32,39)(36,40);(9,11)(10,16)(12,15)(13,19)(14,18)(17,22)(20,21)(23,24)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,39)(38,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,24,21,19)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37);(1,4,5,6)(2,7,8,3)(9,13,14,23)(10,17,15,21)(11,19,18,24)(12,20,16,22)(25,27)(26,28)(29,37)(30,35)(31,33)(32,40)(34,38)(36,39)

Polycyclic presentation:

2^9|a^2=hi,e^2=i,f^2=gh,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, 9625743);