#7658854

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

4458 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7566050);