#9625461

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

4413 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,e^2=gh,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, 9532657);