#8696747

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

4646 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=h,f^2=h,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, 8603943);