#485028

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
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2922 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

705 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=h,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, 392224);