#10360912

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

4713 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)(9,10,14,16)(11,17,19,13)(12,15,20,23)(18,24,21,22)(25,26,32,36)(27,37,40,33)(28,29,30,34)(31,38,39,35);(2,5)(6,7)(9,11,14,19)(10,13,16,17)(12,18,20,21)(15,22,23,24)(25,27,32,40)(26,33,36,37)(28,35,30,38)(29,39,34,31);(1,2)(3,6)(4,5)(7,8)(12,20)(15,23)(18,21)(22,24)(25,28,32,30)(26,34,36,29)(27,38,40,35)(31,33,39,37);(1,3,4,8)(2,6,5,7)(9,11)(10,13)(12,21)(14,19)(15,24)(16,17)(18,20)(22,23)(25,29)(26,28)(27,39)(30,36)(31,40)(32,34)(33,35)(37,38);(9,12)(10,15)(11,18)(13,22)(14,20)(16,23)(17,24)(19,21)(25,30,32,28)(26,34,36,29)(27,38,40,35)(31,37,39,33);(9,13)(10,11)(12,22)(14,17)(15,18)(16,19)(20,24)(21,23)(25,31,32,39)(26,35,36,38)(27,29,40,34)(28,33,30,37)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=hi,e^2=h,f^2=h,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, 10268108);