#7654215

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

4457 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)(11,16)(12,22)(13,23)(14,17)(15,19)(18,20)(21,24)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,39)(38,40);(1,2)(3,6)(4,7)(5,8)(9,11,15,21)(10,16,19,24)(12,20,14,13)(17,23,22,18)(25,27)(26,32)(28,40)(29,38)(30,33)(31,39)(34,35)(36,37);(1,3,5,4)(2,6,8,7)(9,12,15,14)(10,17,19,22)(11,20,21,13)(16,23,24,18)(25,28,31,34)(26,33,36,29)(27,35,39,40)(30,32,38,37);(1,3,5,4)(2,6,8,7)(9,13)(10,18)(11,12)(14,21)(15,20)(16,17)(19,23)(22,24)(25,29,31,33)(26,34,36,28)(27,38,39,30)(32,35,37,40);(9,14,15,12)(10,17,19,22)(11,13,21,20)(16,23,24,18)(25,26)(27,32)(28,33)(29,34)(30,40)(31,36)(35,38)(37,39);(1,4,5,3)(2,7,8,6)(9,10)(11,16)(12,17)(13,18)(14,22)(15,19)(20,23)(21,24)(25,30,31,38)(26,35,36,40)(27,33,39,29)(28,32,34,37)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7561411);