#7868917

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

4563 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)(9,10,15,16)(11,14,19,24)(12,17,21,23)(13,22,20,18)(25,26,31,28)(27,35,37,32)(29,38,40,33)(30,34,36,39);(1,2)(3,7)(4,8)(5,6)(9,11,15,19)(10,14,16,24)(12,20,21,13)(17,18,23,22)(25,27,31,37)(26,32,28,35)(29,36,40,30)(33,39,38,34);(10,16)(11,19)(12,21)(18,22)(26,28)(32,35)(33,38)(34,39);(9,12)(10,17)(11,20)(13,19)(14,18)(15,21)(16,23)(22,24)(25,28,31,26)(27,35,37,32)(29,38,40,33)(30,39,36,34);(1,3,5,4)(2,7,6,8)(9,13,15,20)(10,18,16,22)(11,12,19,21)(14,23,24,17)(25,29)(26,33)(27,36)(28,38)(30,37)(31,40)(32,39)(34,35);(1,4,5,3)(2,8,6,7)(9,14)(10,19)(11,16)(12,18)(13,17)(15,24)(20,23)(21,22)(25,30,31,36)(26,34,28,39)(27,29,37,40)(32,33,35,38)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=hi,e^2=gh,f^2=g,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, 7776113);