#8814034

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

4568 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8721230);