#8720475

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	#2682	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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gi,d^2=gi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8627671);