#7658830

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	#42970	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4224 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)(11,20)(12,18)(13,17)(14,19)(15,16)(21,22)(23,24)(25,26,31,32)(27,30,35,40)(28,37,36,33)(29,34,39,38);(1,2)(3,6)(4,7)(5,8)(10,16)(12,19)(17,20)(21,23)(26,32)(27,35)(28,36)(34,38);(9,11,15,13)(10,17,16,20)(12,21,19,23)(14,22,18,24)(25,27,31,35)(26,30,32,40)(28,29,36,39)(33,38,37,34);(1,3,5,4)(2,6,8,7)(9,12,15,19)(10,18,16,14)(11,21,13,23)(17,24,20,22)(25,28,31,36)(26,33,32,37)(27,29,35,39)(30,38,40,34);(1,3,5,4)(2,6,8,7)(9,13,15,11)(10,17,16,20)(12,23,19,21)(14,22,18,24)(25,29)(26,34)(27,36)(28,35)(30,33)(31,39)(32,38)(37,40);(1,4,5,3)(2,7,8,6)(9,14,15,18)(10,19,16,12)(11,22,13,24)(17,23,20,21)(25,30)(26,35)(27,32)(28,38)(29,37)(31,40)(33,39)(34,36)

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=g,e^2=gi,f^2=gi,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, 7566026);