#7712311

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	#38050	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4375 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)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(26,31)(29,37)(32,38)(33,39);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,26,30,31)(27,32,35,38)(28,33,36,39)(29,34,37,40);(10,14)(11,18)(16,20)(19,24)(26,31)(28,36)(32,38)(34,40);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,27,30,35)(26,32,31,38)(28,34,36,40)(29,33,37,39);(1,4,5,3)(2,8,6,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,28,30,36)(26,33,31,39)(27,34,35,40)(29,32,37,38);(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,29,30,37)(26,34,31,40)(27,33,35,39)(28,32,36,38)

Polycyclic presentation:

2^9|b^2=hi,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

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, 7619507);