#489222

order	512
nilpotency class	4
isoclinism	isoclinic class of #488895
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61409	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

753 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,c^2=h,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396418);