#7673584

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

4311 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,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, 7580780);