#8127507

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

4514 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=h,e^2=h,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, 8034703);