#8533617

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

4772 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)(3,5)(4,7)(6,8)(9,10)(11,16)(12,19)(13,17)(14,15)(18,24)(20,21)(22,23)(25,26,30,32)(27,35,36,31)(28,37,39,33)(29,34,38,40);(2,3)(6,7)(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,24)(22,23)(25,27,30,36)(26,31,32,35)(28,38,39,29)(33,40,37,34);(1,2,5,3)(4,6,8,7)(10,15)(12,16)(17,23)(20,24)(26,32)(31,35)(33,37)(34,40);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,15)(13,21)(14,19)(17,20)(18,22)(23,24)(25,28,30,39)(26,33,32,37)(27,29,36,38)(31,34,35,40);(1,4)(2,6)(3,7)(5,8)(9,12,14,16)(10,11,15,19)(13,20,22,24)(17,18,23,21)(25,26)(27,31)(28,33)(29,34)(30,32)(35,36)(37,39)(38,40);(9,13,14,22)(10,17,15,23)(11,18,19,21)(12,20,16,24)(25,29)(26,34)(27,28)(30,38)(31,33)(32,40)(35,37)(36,39)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440813);