#8125435

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

4672 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,d^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8032631);