#7657888

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

4421 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)(9,10,16,15)(11,20,22,17)(12,23,14,18)(13,19,21,24)(28,36)(31,38)(34,40)(37,39);(1,3,5,8)(2,6,4,7)(9,11)(10,17)(12,13)(14,21)(15,20)(16,22)(18,19)(23,24)(25,26)(27,32)(28,31)(29,33)(30,35)(34,39)(36,38)(37,40);(1,4,5,2)(3,7,8,6)(9,12,16,14)(10,18,15,23)(11,13,22,21)(17,19,20,24)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(9,13,16,21)(10,19,15,24)(11,12,22,14)(17,18,20,23)(25,28,29,36)(26,31,33,38)(27,34,35,40)(30,37,32,39);(9,14,16,12)(10,18,15,23)(11,21,22,13)(17,19,20,24)(25,27)(26,32)(28,34)(29,35)(30,33)(31,39)(36,40)(37,38);(9,15,16,10)(11,20,22,17)(12,23,14,18)(13,24,21,19)(25,26)(27,30)(28,31)(29,33)(32,35)(34,37)(36,38)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=hi,e^2=h,f^2=h,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, 7565084);