#489599

order	512
nilpotency class	4
isoclinism	isoclinic class of #489385
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	#62340	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

709 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)(17,18,23,29)(19,30,33,46)(20,34,38,45)(21,28,22,27)(24,40,43,31)(25,39,44,32)(26,36,41,37)(35,47,48,42);(1,3)(2,7)(4,14)(5,13)(6,12)(8,15)(9,16)(10,11)(17,19)(18,24)(20,35)(21,32)(22,39)(23,33)(25,41)(26,44)(27,42)(28,47)(29,43)(30,34)(31,37)(36,40)(38,48)(45,46);(2,8)(4,5)(7,15)(9,10)(11,16)(13,14)(17,19)(18,25)(20,31)(21,39)(22,32)(23,33)(24,41)(26,43)(27,46)(28,30)(29,44)(34,47)(35,37)(36,48)(38,40)(42,45);(2,9)(3,12)(8,10)(13,14)(17,20,21,36,23,38,22,37)(18,26,27,45,29,41,28,34)(19,31,39,48,33,40,32,35)(24,30,47,25,43,46,42,44)

Polycyclic presentation:

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

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, 396795);