#10020682

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

4501 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9927878);