#8127164

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

4584 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)(9,10,14,18)(11,19,21,15)(12,22,23,16)(13,17,20,24)(25,26)(27,31)(28,35)(29,37)(30,33)(32,38)(34,36)(39,40);(3,7)(6,8)(9,11,14,21)(10,15,18,19)(12,20,23,13)(16,24,22,17)(25,27,30,35)(26,31,33,28)(29,34,38,40)(32,39,37,36);(1,2)(3,6)(4,5)(7,8)(12,23)(13,20)(16,22)(17,24)(25,26,30,33)(27,28,35,31)(29,32,38,37)(34,36,40,39);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,20)(13,21)(14,23)(15,24)(17,19)(18,22)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,20)(10,17,18,24)(11,12,21,23)(15,16,19,22)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,40)(38,39);(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,24)(22,23)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,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, 8034360);