#489343

order	512
nilpotency class	4
isoclinism	isoclinic class of #488920
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	#61489	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

711 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)(3,12)(4,9)(5,10)(6,11)(7,14)(8,13)(15,16)(17,18,23,29)(19,30,31,46)(20,34,38,45)(21,28,22,27)(24,40,42,32)(25,39,44,33)(26,36,41,37)(35,47,48,43);(1,3)(2,7)(4,13)(5,14)(6,15)(8,11)(9,12)(10,16)(17,19)(18,24)(20,35)(21,33)(22,39)(23,31)(25,41)(26,44)(27,43)(28,47)(29,42)(30,34)(32,37)(36,40)(38,48)(45,46);(1,3)(2,8)(4,13)(5,14)(6,15)(7,11)(9,16)(10,12)(17,19)(18,25)(20,32)(21,39)(22,33)(23,31)(24,41)(26,42)(27,46)(28,30)(29,44)(34,47)(35,37)(36,48)(38,40)(43,45);(1,4,6,5)(2,9,11,10)(3,13,15,14)(7,12,8,16)(19,31)(24,42)(25,44)(30,46)(32,40)(33,39)(35,48)(43,47)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396539);