#8094744

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

4261 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)(9,10,15,12)(11,18,22,16)(13,19,14,17)(20,23,21,24)(25,26,31,30)(27,33,28,32)(29,37,38,34)(35,39,36,40);(1,2,5,3)(4,6,8,7)(9,11)(10,16)(12,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,27,31,28)(26,32,30,33)(29,36,38,35)(34,40,37,39);(2,3)(6,7)(9,12,15,10)(11,18,22,16)(13,19,14,17)(20,24,21,23)(25,28,31,27)(26,33,30,32)(29,36,38,35)(34,40,37,39);(9,13,15,14)(10,17,12,19)(11,20,22,21)(16,23,18,24)(25,29)(26,34)(27,35)(28,36)(30,37)(31,38)(32,39)(33,40);(1,3,5,2)(4,7,8,6)(9,11)(10,18)(12,16)(13,20)(14,21)(15,22)(17,24)(19,23);(1,4)(2,6)(3,7)(5,8)(9,14,15,13)(10,19,12,17)(11,21,22,20)(16,24,18,23)(25,30,31,26)(27,33,28,32)(29,37,38,34)(35,40,36,39)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8001940);