#8463198

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4262 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)(3,5)(4,7)(6,8)(9,10)(11,18)(12,21)(13,22)(14,16)(15,17)(19,20)(23,24)(25,26,32,30)(27,37,39,33)(28,35,29,34)(31,36,40,38);(2,3)(6,7)(9,11,15,14)(10,16,17,18)(12,19,22,24)(13,20,21,23)(25,27,32,39)(26,33,30,37)(28,38,29,36)(31,35,40,34);(1,2,5,3)(4,6,8,7)(25,28,32,29)(26,34,30,35)(27,38,39,36)(31,37,40,33);(1,3,5,2)(4,7,8,6)(9,12)(10,13)(11,19)(14,24)(15,22)(16,20)(17,21)(18,23)(25,29,32,28)(26,35,30,34)(27,38,39,36)(31,37,40,33);(1,4)(2,6)(3,7)(5,8)(9,13)(10,12)(11,20)(14,23)(15,21)(16,19)(17,22)(18,24)(25,30,32,26)(27,37,39,33)(28,35,29,34)(31,38,40,36);(9,14,15,11)(10,16,17,18)(12,19,22,24)(13,23,21,20)(25,31,32,40)(26,36,30,38)(27,35,39,34)(28,37,29,33)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8370394);