#507259

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

507 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=efg,c^2=i,d^2=fi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414455);