#7720505

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	#37278	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4542 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)(9,10,15,12)(11,17,13,16)(14,20,22,18)(19,23,21,24)(26,31)(29,40)(32,39)(33,37);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,17)(13,15)(14,19)(18,23)(20,24)(21,22)(25,26,30,31)(27,32,36,39)(28,37,38,33)(29,34,40,35);(10,12)(11,13)(18,20)(19,21)(26,31)(28,38)(32,39)(34,35);(1,4,5,2)(3,7,8,6)(9,12,15,10)(11,17,13,16)(14,20,22,18)(19,24,21,23)(25,27,30,36)(26,32,31,39)(28,35,38,34)(29,33,40,37);(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,28)(26,33)(27,35)(29,39)(30,38)(31,37)(32,40)(34,36);(9,14,15,22)(10,18,12,20)(11,19,13,21)(16,23,17,24)(25,29,30,40)(26,34,31,35)(27,33,36,37)(28,39,38,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=g,e^2=h,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, 7627701);