#483275

order	512
nilpotency class	4
isoclinism	isoclinic class of #482960
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#62475	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

660 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=hi,f^2=gh,h^2=i,b^a=be,c^a=ci,e^a=eg,e^b=eh,f^b=fgh,h^b=hi,d^c=dg

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390471);