#7777927

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

4517 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=ghi,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=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, 7685123);