#10312375

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

4361 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)(4,8)(5,6)(9,10,14,18)(11,15,19,23)(12,16,22,20)(13,17,24,21)(25,26,31,33)(27,34,29,32)(28,38,39,30)(35,37,36,40);(2,6)(7,8)(9,11)(10,15)(12,21)(13,16)(14,19)(17,22)(18,23)(20,24)(25,27,31,29)(26,32,33,34)(28,35,39,36)(30,37,38,40);(1,3,5,4)(2,7,6,8)(11,19)(12,22)(15,23)(16,20)(25,28)(26,30)(27,35)(29,36)(31,39)(32,37)(33,38)(34,40);(1,2,5,6)(3,8,4,7)(9,12)(10,16)(11,17)(13,23)(14,22)(15,24)(18,20)(19,21)(25,28)(26,30)(27,36)(29,35)(31,39)(32,40)(33,38)(34,37);(1,2)(3,7)(4,8)(5,6)(9,13)(10,17)(11,20)(12,15)(14,24)(16,19)(18,21)(22,23)(25,29,31,27)(26,32,33,34)(28,36,39,35)(30,37,38,40);(1,4,5,3)(2,7,6,8)(9,10)(11,15)(12,16)(13,21)(14,18)(17,24)(19,23)(20,22)(25,30)(26,28)(27,37)(29,40)(31,38)(32,35)(33,39)(34,36)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10219571);