#8388123

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

4491 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:

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

Polycyclic presentation:

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

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, 8295319);