#484972

order	512
nilpotency class	4
isoclinism	isoclinic class of #484212
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	C₂²×C₈	C₂⁴
prime radical	#60915	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂²×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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392168);