#7653027

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	#35072	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4371 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)(11,18)(15,22)(19,23)(20,24)(25,26,31,35)(27,36,39,32)(28,33,40,29)(30,34,38,37);(1,2)(3,6)(4,7)(5,8)(9,10,14,17)(11,15,18,22)(12,13,16,21)(19,20,23,24)(25,27)(26,32)(28,34)(29,38)(30,33)(31,39)(35,36)(37,40);(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,28)(26,33)(27,37)(29,35)(30,36)(31,40)(32,38)(34,39);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,29,31,33)(26,28,35,40)(27,38,39,30)(32,34,36,37);(1,4,5,3)(2,7,8,6)(9,12)(10,13)(11,19)(14,16)(15,20)(17,21)(18,23)(22,24)(25,28)(26,33)(27,34)(29,35)(30,32)(31,40)(36,38)(37,39);(9,13)(10,16)(11,20)(12,17)(14,21)(15,23)(18,24)(19,22)(25,30,31,38)(26,34,35,37)(27,33,39,29)(28,36,40,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,e^2=gi,f^2=i,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, 7560223);