#7656571

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
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4165 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=gh,e^2=hi,f^2=hi,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, 7563767);