#507158

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

516 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=egi,c^2=i,d^2=f,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, 414354);