#8186924

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

4430 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8094120);