#7630831

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4686 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,5)(3,8,7,4)(9,10,13,16)(11,14,12,15)(17,18)(19,20)(21,25)(22,36)(23,27)(24,30)(26,37)(28,31)(29,38)(32,43)(33,39)(34,46)(35,41)(40,47)(42,45)(44,48);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,19,23,30)(18,20,27,24)(21,28,35,39)(22,29,37,43)(25,31,41,33)(26,32,36,38)(34,42,47,48)(40,44,46,45);(1,4,6,8)(2,7,5,3)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,33)(26,38)(31,41)(32,36)(34,42)(35,39)(37,43)(40,45)(44,46)(47,48);(9,11,13,12)(10,14,16,15)(17,20,23,24)(18,19,27,30)(21,33,35,31)(22,32,37,38)(25,39,41,28)(26,29,36,43)(34,45,47,44)(40,48,46,42);(1,2,6,5)(3,4,7,8)(17,21,23,35)(18,25,27,41)(19,28,30,39)(20,31,24,33)(22,34,37,47)(26,40,36,46)(29,42,43,48)(32,44,38,45);(1,5,6,2)(3,8,7,4)(9,12,13,11)(10,15,16,14)(17,22)(18,26)(19,29)(20,32)(21,34)(23,37)(24,38)(25,40)(27,36)(28,42)(30,43)(31,44)(33,45)(35,47)(39,48)(41,46)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538027);