#9625448

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

4397 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)(26,31)(28,32)(33,39)(36,37);(1,2)(3,4)(5,6)(7,8)(10,12)(15,17)(16,20)(23,24)(25,26,30,31)(27,32,35,28)(29,33,38,39)(34,37,40,36);(2,6)(3,7)(10,12)(11,18)(16,20)(19,21)(25,27,30,35)(26,32,31,28)(29,34,38,40)(33,37,39,36);(3,7)(4,8)(9,11)(10,15)(12,17)(13,21)(14,18)(16,24)(19,22)(20,23)(25,28)(26,27)(29,37)(30,32)(31,35)(33,40)(34,39)(36,38);(1,3)(2,4)(5,7)(6,8)(9,12,14,10)(11,15,18,17)(13,20,22,16)(19,23,21,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,37)(36,39)(38,40);(1,4)(2,3)(5,8)(6,7)(9,13,14,22)(10,16,12,20)(11,19,18,21)(15,23,17,24)(25,29,30,38)(26,33,31,39)(27,34,35,40)(28,36,32,37)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,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, 9532644);