#488659

order	512
nilpotency class	4
isoclinism	isoclinic class of #488531
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	#3191	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#433 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

659 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 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)(34,40)(36,38)(37,49)(41,56)(42,60)(43,55)(44,47)(45,61)(48,52)(50,53)(51,63)(54,58)(57,62)(59,64);(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)(33,34,39,43)(35,41,46,58)(36,42,38,53)(37,50,52,60)(40,48,55,49)(44,57,47,59)(45,62,51,64)(54,61,56,63);(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)(33,35)(34,41)(36,47)(37,51)(38,44)(39,46)(40,54)(42,59)(43,58)(45,52)(48,61)(49,63)(50,64)(53,57)(55,56)(60,62);(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)(33,36,39,38)(34,42,43,53)(35,44,46,47)(37,48,52,49)(40,50,55,60)(41,57,58,59)(45,61,51,63)(54,62,56,64)

Polycyclic presentation:

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

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, 395855);