#10323715

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

4377 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=g,e^2=gh,f^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, 10230911);