#489591

order	512
nilpotency class	4
isoclinism	isoclinic class of #489379
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	#62324	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

710 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)(3,11)(4,10)(5,8)(6,9)(7,13)(12,15)(14,16)(17,18,22,28)(19,24,32,41)(20,23,35,38)(21,36,34,27)(25,33,26,43)(29,44,30,37)(31,48,46,40)(39,45,42,47);(1,3)(2,7)(4,14)(5,13)(6,12)(8,15)(9,16)(10,11)(18,23)(20,33)(21,34)(24,37)(25,36)(26,27)(28,38)(29,45)(30,47)(31,46)(35,43)(39,40)(41,44)(42,48);(2,8)(4,5)(7,15)(9,10)(11,16)(13,14)(17,19)(18,24)(20,30)(21,31)(22,32)(23,37)(25,42)(26,39)(27,40)(28,41)(29,35)(33,47)(34,46)(36,48)(38,44)(43,45);(2,9)(3,12)(8,10)(13,14)(17,20,34,43,22,35,21,33)(18,25,36,23,28,26,27,38)(19,29,46,47,32,30,31,45)(24,39,48,44,41,42,40,37)

Polycyclic presentation:

2^9|a^2=i,d^2=ghi,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,d^c=di,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, 396787);