#7709429

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

4488 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)(7,8)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(26,32)(30,40)(33,35)(37,39);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,26,31,32)(27,35,28,33)(29,30,34,40)(36,37,38,39);(1,3,5,4)(2,7,6,8)(10,14)(11,18)(16,20)(19,24)(25,27,31,28)(26,33,32,35)(29,38,34,36)(30,37,40,39);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,28,31,27)(26,33,32,35)(29,38,34,36)(30,39,40,37);(1,4,5,3)(2,8,6,7)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,29,31,34)(26,30,32,40)(27,36,28,38)(33,39,35,37);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,30)(26,34)(27,37)(28,39)(29,32)(31,40)(33,36)(35,38)

Polycyclic presentation:

2^9|b^2=hi,c^2=g,d^2=g,e^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7616625);