#489462

order	512
nilpotency class	4
isoclinism	isoclinic class of #489365
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	#62475	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

674 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396658);