#7656857

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	#392	C₂³
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)(11,22)(12,23)(13,19)(14,18)(15,20)(16,21)(17,24)(25,26,31,35)(27,34,36,30)(28,32,39,40)(29,37,38,33);(1,2)(3,6)(4,7)(5,8)(9,11)(10,17)(12,13)(14,20)(15,23)(16,24)(18,19)(21,22)(25,27)(26,30)(28,37)(29,32)(31,36)(33,39)(34,35)(38,40);(9,12,16,14)(10,18,21,23)(11,13,24,20)(15,17,19,22)(25,28,31,39)(26,32,35,40)(27,33,36,37)(29,30,38,34);(1,3,5,4)(2,6,8,7)(9,13,16,20)(10,19,21,15)(11,12,24,14)(17,18,22,23)(25,29,31,38)(26,33,35,37)(27,32,36,40)(28,30,39,34);(1,3,5,4)(2,6,8,7)(9,14,16,12)(10,18,21,23)(11,20,24,13)(15,17,19,22)(25,30)(26,27)(28,38)(29,39)(31,34)(32,33)(35,36)(37,40);(1,4,5,3)(2,7,8,6)(9,15,16,19)(10,20,21,13)(11,23,24,18)(12,17,14,22)(25,27)(26,34)(28,33)(29,32)(30,35)(31,36)(37,39)(38,40)

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,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, 7564053);