#7987268

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
prime radical	#49897	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4436 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,d^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7894464);