#7657031

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

4345 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)(11,17)(12,16)(13,18)(14,19)(15,22)(20,21)(23,24)(28,36)(33,38)(35,37)(39,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,19)(18,24)(25,26,31,30)(27,34,29,32)(28,33,36,38)(35,40,37,39);(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,19)(18,24)(25,27,31,29)(26,32,30,34)(28,35,36,37)(33,39,38,40);(1,3,5,4)(2,7,6,8)(9,12,14,18)(10,16,19,13)(11,20,22,24)(15,23,17,21)(25,28)(26,33)(27,35)(29,37)(30,38)(31,36)(32,39)(34,40);(1,4,5,3)(2,8,6,7)(9,11)(10,17)(12,20)(13,23)(14,22)(15,19)(16,21)(18,24)(25,29,31,27)(26,32,30,34)(28,37,36,35)(33,39,38,40);(9,13,14,16)(10,18,19,12)(11,21,22,23)(15,24,17,20)(25,30,31,26)(27,34,29,32)(28,38,36,33)(35,40,37,39)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=g,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, 7564227);