#8750177

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

4501 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=gi,d^2=gh,e^2=gh,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, 8657373);