#8750101

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

4235 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=hi,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8657297);