#7659680

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

4561 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,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, 7566876);