#8174299

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

4510 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gi,d^2=h,e^2=h,f^2=gi,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^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, 8081495);