#7657127

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

4379 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)(12,20)(17,22)(19,21)(23,24)(25,26)(27,35)(28,32)(29,33)(30,34)(31,38)(36,39)(37,40);(1,2,5,6)(3,7,4,8)(9,10,15,14)(11,18,13,16)(12,17,20,22)(19,24,21,23)(25,27)(26,31)(28,36)(29,37)(30,38)(32,33)(34,35)(39,40);(9,11,15,13)(10,16,14,18)(12,19,20,21)(17,23,22,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,33)(34,35)(39,40);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,19)(13,21)(14,22)(15,20)(16,23)(18,24)(25,28,30,37)(26,32,34,40)(27,36,38,29)(31,33,35,39);(1,4,5,3)(2,8,6,7)(9,13,15,11)(10,16,14,18)(12,21,20,19)(17,23,22,24)(25,29,30,36)(26,33,34,39)(27,37,38,28)(31,32,35,40);(9,14,15,10)(11,18,13,16)(12,22,20,17)(19,24,21,23)(25,26)(27,31)(28,32)(29,39)(30,34)(33,36)(35,38)(37,40)

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=g,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7564323);