#9706775

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

4373 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)(11,18)(12,17)(13,22)(14,15)(16,21)(19,20)(23,24)(25,26)(27,28)(29,37)(30,32)(31,39)(33,35)(34,38)(36,40);(1,2)(3,8)(4,6)(5,7)(10,15)(12,22)(16,18)(19,24)(25,27,30,35)(26,28,32,33)(29,34,39,40)(31,36,37,38);(1,3,5,4)(2,6,7,8)(9,11,14,21)(10,16,15,18)(12,19,22,24)(13,20,17,23)(25,26,30,32)(27,33,35,28)(29,31,39,37)(34,38,40,36);(9,11)(10,16)(12,19)(13,20)(14,21)(15,18)(17,23)(22,24)(25,26)(27,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40);(1,4,5,3)(2,6,7,8)(9,12,14,22)(10,17,15,13)(11,19,21,24)(16,23,18,20)(25,28)(26,27)(29,38)(30,33)(31,40)(32,35)(34,37)(36,39);(1,2)(3,6)(4,8)(5,7)(9,13,14,17)(10,12,15,22)(11,20,21,23)(16,19,18,24)(25,29)(26,31)(27,34)(28,36)(30,39)(32,37)(33,38)(35,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,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, 9613971);