#8108752

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

4492 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:

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

Polycyclic presentation:

2^9|c^2=gi,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015948);