#7656801

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

4463 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=g,e^2=gh,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, 7563997);