#8323165

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

4460 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:

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

Polycyclic presentation:

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

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, 8230361);