#7674484

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	#34643	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4645 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=gi,e^2=ghi,f^2=gh,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, 7581680);