#500163

order	512
nilpotency class	4
isoclinism	isoclinic class of #500160
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24033
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61091	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

934 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307076, #72307089, #72307112, …
maximal normal subgroup of	#72307076, #72307089, #72307112, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407359);