#8701568

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4081 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=gi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8608764);