#7657075

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

4167 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)(12,20)(17,22)(19,21)(23,24)(25,26,32,31)(27,36,30,33)(28,38,37,34)(29,35,39,40);(1,3,5,8)(2,6,4,7)(9,10,15,14)(11,18,13,16)(12,17,20,22)(19,24,21,23)(25,27,32,30)(26,33,31,36)(28,29,37,39)(34,35,38,40);(9,11,15,13)(10,16,14,18)(12,19,20,21)(17,23,22,24)(25,28)(26,34)(27,29)(30,39)(31,38)(32,37)(33,35)(36,40);(1,4,5,2)(3,7,8,6)(9,12)(10,17)(11,19)(13,21)(14,22)(15,20)(16,23)(18,24)(25,29,32,39)(26,35,31,40)(27,37,30,28)(33,38,36,34);(9,13,15,11)(10,16,14,18)(12,21,20,19)(17,23,22,24)(25,30,32,27)(26,33,31,36)(28,39,37,29)(34,35,38,40);(9,14,15,10)(11,18,13,16)(12,22,20,17)(19,24,21,23)(25,31,32,26)(27,36,30,33)(28,38,37,34)(29,40,39,35)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=g,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, 7564271);