#8745807

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

4527 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,d^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653003);