#9674962

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

4687 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=h,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, 9582158);