#489225

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	#61406	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

727 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)(19,25)(20,26)(21,34)(22,38)(23,28)(24,30)(27,36)(29,42)(31,44)(32,43)(33,45)(35,39)(37,48)(40,47)(41,46);(4,13)(7,12)(9,16)(11,14)(17,19)(18,24)(20,29)(21,35)(22,32)(23,31)(25,34)(26,39)(27,43)(28,41)(30,36)(33,46)(37,47)(38,40)(42,45)(44,48);(1,3,5,10)(2,6,8,15)(4,14,13,11)(7,16,12,9)(17,19)(18,25)(20,32)(21,30)(22,29)(23,31)(24,34)(26,43)(27,39)(28,44)(33,47)(35,36)(37,46)(38,42)(40,45)(41,48);(17,20,23,22)(18,26,28,38)(19,29,31,32)(21,33,37,36)(24,39,41,40)(25,42,44,43)(27,34,45,48)(30,35,46,47)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=i,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, 396421);