#9625393

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

4412 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)(11,17,18,15)(13,20,22,16)(19,23,21,24)(25,26,31,35)(27,33,28,32)(29,34,40,30)(36,37,38,39);(1,2)(3,4)(5,6)(7,8)(10,12)(15,17)(16,20)(23,24)(25,27,31,28)(26,32,35,33)(29,36,40,38)(30,37,34,39);(2,6)(3,7)(10,12)(11,18)(16,20)(19,21)(25,27,31,28)(26,33,35,32)(29,38,40,36)(30,37,34,39);(3,7)(4,8)(9,11,14,18)(10,15,12,17)(13,21,22,19)(16,24,20,23)(25,28,31,27)(26,33,35,32)(29,36,40,38)(30,37,34,39);(1,3)(2,4)(5,7)(6,8)(9,12,14,10)(11,15,18,17)(13,20,22,16)(19,23,21,24)(25,29)(26,34)(27,36)(28,38)(30,35)(31,40)(32,39)(33,37);(1,4)(2,3)(5,8)(6,7)(9,13,14,22)(10,16,12,20)(11,19,18,21)(15,23,17,24)(25,30,31,34)(26,29,35,40)(27,37,28,39)(32,36,33,38)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=g,e^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,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9532589);