#7888610

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

4313 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)(9,10)(11,19)(12,22)(13,17)(14,18)(15,21)(16,23)(20,24)(25,26,32,38)(27,33,29,35)(28,36,30,34)(31,37,40,39);(1,2)(3,7)(4,8)(5,6)(9,11,14,21)(10,15,18,19)(12,20,23,13)(16,24,22,17)(25,27,32,29)(26,33,38,35)(28,39,30,37)(31,34,40,36);(9,10,14,18)(11,15,21,19)(12,16,23,22)(13,17,20,24)(25,28,32,30)(26,34,38,36)(27,37,29,39)(31,35,40,33);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,24)(22,23)(25,29,32,27)(26,35,38,33)(28,37,30,39)(31,36,40,34);(9,12)(10,16)(11,20)(13,21)(14,23)(15,24)(17,19)(18,22)(25,30,32,28)(26,36,38,34)(27,37,29,39)(31,35,40,33);(1,4,5,3)(2,8,6,7)(9,13,14,20)(10,17,18,24)(11,12,21,23)(15,16,19,22)(25,31,32,40)(26,37,38,39)(27,34,29,36)(28,35,30,33)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=gh,e^2=i,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7795806);