#7647720

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	#35189	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4444 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=h,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,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, 7554916);