#10422825

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

4600 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)(3,5)(4,6)(7,8)(10,11)(12,19)(13,21)(22,23)(25,26,31,35)(27,34,38,32)(28,30,29,33)(36,37,39,40);(2,3)(6,7)(9,10,14,11)(12,17,19,15)(13,16,21,18)(20,23,24,22)(25,27)(26,32)(28,39)(29,36)(30,37)(31,38)(33,40)(34,35);(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(25,28,31,29)(26,33,35,30)(27,36,38,39)(32,37,34,40);(2,3)(4,8)(9,11,14,10)(12,17,19,15)(13,16,21,18)(20,22,24,23)(25,29,31,28)(26,30,35,33)(27,36,38,39)(32,37,34,40);(1,3,5,2)(4,7,8,6)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,27)(26,34)(28,36)(29,39)(30,37)(31,38)(32,35)(33,40);(1,4,5,8)(2,6,3,7)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30)(26,28)(27,37)(29,35)(31,33)(32,39)(34,36)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=h,e^2=gh,f^2=gh,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, 10330021);