#8258052

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

4328 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8165248);