#8137323

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

4377 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)(6,7)(12,20)(16,23)(18,24)(21,22)(25,26,31,36)(27,37,38,32)(28,34,29,33)(30,39,40,35);(1,2,5,4)(3,6,8,7)(9,10)(11,15)(12,16)(13,17)(14,19)(18,22)(20,23)(21,24)(25,27,31,38)(26,32,36,37)(28,35,29,39)(30,34,40,33);(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,28,31,29)(26,33,36,34)(27,35,38,39)(30,37,40,32);(1,2,5,4)(3,7,8,6)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,29,31,28)(26,34,36,33)(27,35,38,39)(30,37,40,32);(1,3,5,8)(2,6,4,7)(9,10)(11,14)(12,16)(13,17)(15,19)(18,21)(20,23)(22,24)(25,30)(26,35)(27,34)(28,37)(29,32)(31,40)(33,38)(36,39);(1,4,5,2)(3,7,8,6)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,26)(27,32)(28,33)(29,34)(30,35)(31,36)(37,38)(39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=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, 8044519);