#7650875

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

4403 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,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, 7558071);