#7657251

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	#42064	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4230 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)(11,19,21,16)(12,17,20,22)(13,18,23,24)(25,26,29,30)(27,33,35,31)(28,32,36,37)(34,39,40,38);(1,2)(3,6)(4,7)(5,8)(9,11,15,21)(10,16,14,19)(12,13,20,23)(17,24,22,18)(26,30)(31,33)(32,37)(38,39);(9,11,15,21)(10,16,14,19)(12,13,20,23)(17,24,22,18)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,3,5,4)(2,6,8,7)(9,12,15,20)(10,17,14,22)(11,13,21,23)(16,24,19,18)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,4,5,3)(2,7,8,6)(9,13)(10,18)(11,20)(12,21)(14,24)(15,23)(16,17)(19,22);(9,14,15,10)(11,19,21,16)(12,22,20,17)(13,18,23,24)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=hi,d^2=g,e^2=gh,f^2=h,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, 7564447);