#7652344

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	#40285	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4290 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=i,e^2=gi,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, 7559540);