#10460712

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

4610 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10367908);