#496615

order	512
nilpotency class	4
isoclinism	isoclinic class of #496611
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#21888
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60717	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

739 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=ehi,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,f^b=fi,g^b=gi,e^c=ei,f^c=fh,h^c=hi,f^d=fi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403811);