#7644320

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	#42971	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4360 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7551516);