#7646772

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

4444 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7553968);