#7630661

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4786 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:

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

Polycyclic presentation:

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

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, 7537857);