#8612229

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

4514 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8519425);