#7652776

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	#42977	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=gi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

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, 7559972);