#7655314

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

4438 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)(9,10,15,14)(11,19,22,16)(12,18,13,17)(20,23,21,24)(25,26)(27,36)(28,29)(30,34)(31,33)(32,35)(37,38)(39,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,16)(12,20)(13,21)(14,19)(15,22)(17,23)(18,24)(25,27,32,31)(26,33,35,36)(28,39,30,37)(29,38,34,40);(9,12,15,13)(10,17,14,18)(11,20,22,21)(16,23,19,24)(25,28,32,30)(26,29,35,34)(27,37,31,39)(33,40,36,38);(1,3,5,4)(2,6,8,7)(9,13,15,12)(10,18,14,17)(11,21,22,20)(16,24,19,23)(25,29,32,34)(26,28,35,30)(27,38,31,40)(33,39,36,37);(1,3,5,4)(2,6,8,7)(9,11)(10,19)(12,20)(13,21)(14,16)(15,22)(17,24)(18,23)(25,30,32,28)(26,34,35,29)(27,37,31,39)(33,40,36,38);(1,4,5,3)(2,7,8,6)(9,14,15,10)(11,19,22,16)(12,18,13,17)(20,24,21,23)(25,31,32,27)(26,33,35,36)(28,39,30,37)(29,40,34,38)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,d^2=g,e^2=gh,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, 7562510);