#483549

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#449	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

442 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,20,25,21,26,30,38,22,24,31,36,32,37,42,46)(19,23,27,34,28,35,39,45,29,33,40,43,41,44,47,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13);(17,19,20,27,21,28,30,39,22,29,31,40,32,41,42,47)(18,23,25,34,26,35,38,45,24,33,36,43,37,44,46,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(18,24)(23,33)(25,36)(26,37)(34,43)(35,44)(38,46)(45,48)

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=g,e^2=g,f^2=h,h^2=i,b^a=be,d^a=di,e^a=eg,e^b=eg

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390745);