#9706723

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

4325 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ghi,d^2=gh,e^2=gi,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, 9613919);