#10332968

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

4555 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)(9,10,14,12)(11,15,19,17)(13,20,21,16)(18,23,24,22)(25,26,30,32)(27,35,37,31)(28,38,39,33)(29,34,40,36);(9,10,14,12)(11,17,19,15)(13,20,21,16)(18,22,24,23)(25,27,30,37)(26,31,32,35)(28,34,39,36)(29,33,40,38);(1,2)(3,6)(4,5)(7,8)(10,12)(15,17)(16,20)(22,23)(26,32)(29,40)(31,35)(33,38);(3,7)(6,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,28,30,39)(26,33,32,38)(27,36,37,34)(29,35,40,31);(1,3,4,7)(2,6,5,8)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,29)(26,34)(27,33)(28,35)(30,40)(31,39)(32,36)(37,38);(1,2)(3,6)(4,5)(7,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,27)(26,31)(28,36)(29,33)(30,37)(32,35)(34,39)(38,40)

Polycyclic presentation:

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

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, 10240164);