#8749695

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

4329 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)(3,7)(4,8)(5,6)(9,10)(11,17)(12,15)(13,16)(14,19)(18,24)(20,22)(21,23)(27,29)(28,32)(35,37)(36,38);(2,6)(3,8)(9,10,13,16)(11,14,19,17)(12,15,22,20)(18,23,21,24)(25,26,31,34)(27,28,29,32)(30,33,39,40)(35,36,37,38);(1,3)(2,4)(5,8)(6,7)(9,10,13,16)(11,14,19,17)(12,15,22,20)(18,23,21,24)(25,27,31,29)(26,32,34,28)(30,35,39,37)(33,38,40,36);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,20)(13,16)(15,22)(17,19)(18,24)(21,23)(25,28,31,32)(26,27,34,29)(30,38,39,36)(33,37,40,35);(1,4)(2,3)(5,7)(6,8)(9,11)(10,14)(12,21)(13,19)(15,24)(16,17)(18,22)(20,23)(25,29,31,27)(26,32,34,28)(30,35,39,37)(33,36,40,38);(9,12,13,22)(10,15,16,20)(11,18,19,21)(14,23,17,24)(25,30,31,39)(26,33,34,40)(27,35,29,37)(28,36,32,38)

Polycyclic presentation:

2^9|b^2=gi,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, 8656891);