#9848374

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

4426 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)(27,29)(28,31)(30,36)(32,34)(33,38)(35,39)(37,40);(2,3)(6,7)(9,10,14,11)(12,15,20,17)(13,16,22,18)(19,23,21,24)(25,27)(26,32)(28,29)(30,35)(31,34)(33,39)(36,37)(38,40);(1,2,5,3)(4,6,8,7)(10,11)(15,17)(16,18)(23,24)(26,28)(27,34)(33,36)(35,40);(9,11,14,10)(12,17,20,15)(13,18,22,16)(19,24,21,23)(25,28,31,26)(27,29,34,32)(30,36,38,33)(35,37,40,39);(1,3,5,2)(4,6,8,7)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,29)(26,27)(28,34)(30,37)(31,32)(33,35)(36,40)(38,39);(1,4)(2,6)(3,7)(5,8)(9,13)(10,16)(11,18)(12,19)(14,22)(15,23)(17,24)(20,21)(25,30,31,38)(26,33,28,36)(27,35,34,40)(29,37,32,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9755570);