#8445741

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

4380 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8352937);