#9844106

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

4465 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=gh,e^2=gi,f^2=gh,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, 9751302);