#7859273

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) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7766469);