#7891230

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

4520 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=hi,e^2=gi,f^2=ghi,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, 7798426);