#7657234

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	#41996	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4226 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)(11,21)(12,17)(13,18)(14,19)(15,20)(16,22)(23,24)(25,26)(27,29)(28,36)(30,31)(32,34)(33,38)(35,39)(37,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,22)(10,16,20,21)(12,13,19,23)(14,18,17,24)(26,31)(27,34)(33,36)(35,40);(9,11,15,22)(10,16,20,21)(12,13,19,23)(14,18,17,24)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,37)(38,40);(1,3,5,4)(2,7,6,8)(9,12,15,19)(10,17,20,14)(11,13,22,23)(16,24,21,18)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(1,3,5,4)(2,7,6,8)(9,13)(10,18)(11,19)(12,22)(14,21)(15,23)(16,17)(20,24);(1,4,5,3)(2,8,6,7)(9,14,15,17)(10,19,20,12)(11,18,22,24)(13,16,23,21)(25,29,30,32)(26,27,31,34)(28,37,38,39)(33,35,36,40)

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,d^2=g,e^2=gh,f^2=g,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, 7564430);