#7656542

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	#42798	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4296 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)(12,17,21,23)(13,18,20,24)(25,26,32,31)(27,36,34,29)(28,33,39,37)(30,35,40,38);(1,2)(3,6)(4,7)(5,8)(25,27)(26,29)(28,38)(30,37)(31,36)(32,34)(33,40)(35,39);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,20)(13,21)(14,19)(15,22)(17,18)(23,24)(25,28,32,39)(26,33,31,37)(27,35,34,38)(29,30,36,40);(1,3,5,4)(2,6,8,7)(9,12,15,21)(10,17,14,23)(11,20,22,13)(16,18,19,24)(25,29,32,36)(26,34,31,27)(28,30,39,40)(33,38,37,35);(1,4,5,3)(2,7,8,6)(9,13,15,20)(10,18,14,24)(11,21,22,12)(16,17,19,23)(25,30)(26,35)(27,37)(28,36)(29,39)(31,38)(32,40)(33,34);(9,14,15,10)(11,19,22,16)(12,23,21,17)(13,18,20,24)(25,31,32,26)(27,36,34,29)(28,37,39,33)(30,35,40,38)

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=gh,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7563738);