#10463522

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

4519 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^2=g,e^2=g,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, 10370718);