#501151

order	512
nilpotency class	4
isoclinism	isoclinic class of #501150
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24404
commutator, Frattini, cosocle	C₂×#41	C₂⁴
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1054 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,6,13,8,15,7,14)(3,16,19,47,20,44,32,62)(4,21,25,41,23,40,26,10)(5,12,30,45,29,11,31,43)(9,33,35,18,37,52,36,27)(17,48,22,56,49,64,53,46)(24,55,59,39,28,61,60,42)(34,50,63,58,54,51,38,57);(1,3,8,20)(2,9,15,37)(4,22,23,53)(5,27,29,18)(6,19,7,32)(10,38,41,63)(11,16,12,44)(13,36,14,35)(17,25,49,26)(21,54,40,34)(24,58,28,57)(30,33,31,52)(39,64,42,48)(43,62,45,47)(46,61,56,55)(50,59,51,60);(1,4)(2,10)(3,17)(5,28)(6,26)(7,25)(8,23)(9,34)(11,42)(12,39)(13,40)(14,21)(15,41)(16,46)(18,51)(19,53)(20,49)(22,32)(24,29)(27,50)(30,59)(31,60)(33,57)(35,38)(36,63)(37,54)(43,61)(44,56)(45,55)(47,64)(48,62)(52,58);(2,11)(4,23)(5,29)(9,16)(10,39)(12,15)(13,45)(14,43)(17,49)(18,27)(21,55)(22,53)(25,26)(30,31)(33,52)(34,56)(35,62)(36,47)(37,44)(38,64)(40,61)(41,42)(46,54)(48,63)

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,d^c=di,e^c=eh,f^c=fi,g^c=gi,e^d=ei,f^d=fhi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408347);