#488667

order	512
nilpotency class	4
isoclinism	isoclinic class of #488539
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	#3203	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2922 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

633 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 96
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 96 points:

(1,2,6,10)(3,7,13,20)(4,8,15,22)(5,16,17,9)(11,23,25,18)(12,26,27,19)(14,28,29,21)(24,30,32,31)(33,34,40,43)(35,47,52,77)(36,53,56,75)(37,57,62,74)(38,46,39,45)(41,64,66,49)(42,63,69,55)(44,60,71,61)(48,79,82,96)(50,72,51,78)(54,73,59,70)(58,67,84,68)(65,88,93,91)(76,85,94,81)(80,86,83,95)(87,92,90,89);(1,3)(2,7)(4,11)(5,12)(6,13)(8,18)(9,19)(10,20)(14,24)(15,25)(16,26)(17,27)(21,30)(22,23)(28,31)(29,32)(33,35,40,52)(34,41,43,66)(36,48,56,82)(37,58,62,84)(38,51,39,50)(42,65,69,93)(44,72,71,78)(45,68,46,67)(47,74,77,57)(49,60,64,61)(53,86,75,95)(54,88,59,91)(55,85,63,81)(70,96,73,79)(76,87,94,90)(80,92,83,89);(1,4)(2,8)(3,11)(5,14)(6,15)(7,18)(9,21)(10,22)(12,24)(13,25)(16,28)(17,29)(19,30)(20,23)(26,31)(27,32)(33,36)(34,42)(35,48)(37,59)(38,63)(39,55)(40,56)(41,65)(43,69)(44,73)(45,75)(46,53)(47,76)(49,83)(50,85)(51,81)(52,82)(54,62)(57,90)(58,91)(60,89)(61,92)(64,80)(66,93)(67,86)(68,95)(70,71)(72,79)(74,87)(77,94)(78,96)(84,88);(1,5,6,17)(2,9,10,16)(3,12,13,27)(4,14,15,29)(7,19,20,26)(8,21,22,28)(11,24,25,32)(18,30,23,31)(34,43)(41,66)(42,69)(44,71)(45,46)(47,77)(53,75)(57,74)(65,93)(67,68)(70,73)(72,78)(76,94)(79,96)(86,95)(87,90)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=hi,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,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, 395863);