#485038

order	512
nilpotency class	4
isoclinism	isoclinic class of #484217
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	#61014	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2922 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

707 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:

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

Polycyclic presentation:

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

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, 392234);