#8118773

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

4970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 8025969);