#8301702

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
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4293 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=gh,e^2=gi,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, 8208898);