#8107498

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

4477 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014694);