#9718536

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

4553 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)(3,4)(5,7)(6,8)(9,10,14,15)(11,16,18,12)(13,21,23,17)(19,20,24,22)(25,26)(27,33)(28,32)(29,34)(30,31)(35,40)(36,37)(38,39);(1,3)(2,6)(4,7)(5,8)(10,15)(12,16)(17,21)(20,22)(25,27)(26,29)(28,36)(30,34)(31,33)(32,40)(35,38)(37,39);(10,15)(11,18)(17,21)(19,24)(26,31)(27,34)(32,39)(35,36);(9,11)(10,16)(12,15)(13,19)(14,18)(17,22)(20,21)(23,24)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,38)(39,40);(1,4,5,6)(2,3,7,8)(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,24,21,19)(25,28,30,38)(26,32,31,39)(27,35,34,36)(29,37,33,40);(1,3)(2,4)(5,8)(6,7)(9,13,14,23)(10,17,15,21)(11,19,18,24)(12,20,16,22)(25,29)(26,27)(28,37)(30,33)(31,34)(32,35)(36,39)(38,40)

Polycyclic presentation:

2^9|a^2=hi,e^2=g,f^2=hi,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, 9625732);