#7719872

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

4480 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7627068);