#8127205

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

4717 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)(11,15)(12,19)(13,21)(14,17)(16,22)(18,20)(23,24)(25,26)(27,36)(28,29)(30,33)(31,32)(34,35)(37,40)(38,39);(3,7)(6,8)(9,11,14,19)(10,15,17,12)(13,18,22,24)(16,23,21,20)(26,32)(28,35)(33,36)(37,39);(1,2)(3,6)(4,5)(7,8)(9,10,14,17)(11,12,19,15)(13,16,22,21)(18,20,24,23)(25,27,31,30)(26,33,32,36)(28,39,35,37)(29,40,34,38);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,16)(14,17)(15,19)(18,20)(21,22)(23,24)(25,28,31,35)(26,34,32,29)(27,37,30,39)(33,40,36,38);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,24)(22,23)(25,29,31,34)(26,35,32,28)(27,38,30,40)(33,39,36,37);(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,21)(19,24)(25,30,31,27)(26,36,32,33)(28,39,35,37)(29,40,34,38)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8034401);