#10320428

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

4273 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=hi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,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, 10227624);