#8127161

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

4792 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)(12,20)(13,18)(14,15)(16,21)(19,22)(23,24)(25,26)(27,31)(28,35)(29,37)(30,33)(32,38)(34,36)(39,40);(3,7)(6,8)(10,15)(11,18)(16,20)(19,24)(25,27,30,35)(26,31,33,28)(29,34,38,40)(32,39,37,36);(1,2)(3,6)(4,5)(7,8)(10,15)(11,18)(12,21)(22,23)(25,26,30,33)(27,28,35,31)(29,32,38,37)(34,36,40,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,19)(14,18)(15,17)(16,22)(20,23)(21,24)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,21)(10,16,15,20)(11,19,18,24)(13,22,17,23)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,40)(38,39);(9,13)(10,11)(12,22)(14,17)(15,18)(16,19)(20,24)(21,23)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^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, 8034357);