#7656938

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	#42234	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4233 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)(9,10,16,20)(11,21,14,17)(12,23,22,18)(13,15,19,24)(25,26,31,30)(27,34,29,32)(28,36,38,33)(35,39,37,40);(1,2)(3,6)(4,7)(5,8)(9,11,16,14)(10,17,20,21)(12,13,22,19)(15,23,24,18)(25,27,31,29)(26,32,30,34)(28,37,38,35)(33,40,36,39);(1,3,5,4)(2,6,8,7)(9,12)(10,18)(11,13)(14,19)(15,17)(16,22)(20,23)(21,24)(25,28,31,38)(26,33,30,36)(27,35,29,37)(32,39,34,40);(1,3,5,4)(2,6,8,7)(9,13,16,19)(10,15,20,24)(11,22,14,12)(17,23,21,18)(25,29,31,27)(26,34,30,32)(28,37,38,35)(33,40,36,39);(1,4,5,3)(2,7,8,6)(9,14,16,11)(10,17,20,21)(12,19,22,13)(15,23,24,18);(9,15)(10,19)(11,23)(12,17)(13,20)(14,18)(16,24)(21,22)(25,30,31,26)(27,34,29,32)(28,36,38,33)(35,40,37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=g,e^2=gi,f^2=i,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, 7564134);