#10220709

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

4467 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10127905);