#7655177

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	#42235	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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7562373);