#7712733

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	#2615	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4396 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=gh,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, 7619929);