#8617824

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

4653 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)(6,8)(9,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,23)(22,24)(25,26,30,33)(27,34,29,31)(28,32,37,40)(35,39,36,38);(3,7)(6,8)(9,11,15,17)(10,12,13,20)(14,18,23,24)(16,19,21,22)(25,27,30,29)(26,31,33,34)(28,36,37,35)(32,39,40,38);(1,2)(3,6)(4,5)(7,8)(10,13)(11,17)(16,21)(18,24)(28,37)(32,40)(35,36)(38,39);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,17)(14,22)(15,20)(16,24)(18,21)(19,23)(25,26)(27,31)(28,32)(29,34)(30,33)(35,38)(36,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,10)(11,12,17,20)(14,21,23,16)(18,19,24,22)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,40)(34,39);(9,14)(10,16)(11,18)(12,19)(13,21)(15,23)(17,24)(20,22)(25,29,30,27)(26,31,33,34)(28,36,37,35)(32,38,40,39)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8525020);