#7712677

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	#35259	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4474 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)(10,16)(14,18)(17,20)(22,24)(26,29)(27,30)(33,37)(35,36);(1,2)(3,7)(4,8)(5,6)(9,10)(11,20)(12,17)(13,18)(14,19)(15,16)(21,24)(22,23)(25,26,31,29)(27,34,30,32)(28,33,38,37)(35,40,36,39);(1,3,5,4)(2,7,6,8)(9,11,15,12)(10,17,16,20)(13,23,19,21)(14,22,18,24)(25,27,31,30)(26,32,29,34)(28,36,38,35)(33,40,37,39);(1,4,5,3)(2,8,6,7)(9,12,15,11)(10,17,16,20)(13,23,19,21)(14,24,18,22)(25,28,31,38)(26,33,29,37)(27,35,30,36)(32,39,34,40);(9,13,15,19)(10,18,16,14)(11,21,12,23)(17,22,20,24)(25,29,31,26)(27,32,30,34)(28,37,38,33)(35,39,36,40);(9,14,15,18)(10,19,16,13)(11,22,12,24)(17,21,20,23)(25,30,31,27)(26,32,29,34)(28,36,38,35)(33,39,37,40)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=g,e^2=i,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, 7619873);