#8583428

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

4451 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,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, 8490624);