#10324552

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

4911 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,e^2=g,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, 10231748);