#10444995

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

4509 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=ghi,d^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

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, 10352191);