#8904555

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

4461 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8811751);