#494760

order	512
nilpotency class	4
isoclinism	isoclinic class of #494749
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1898
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#585 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1957 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280647, #72281946, #72282134, …
maximal normal subgroup of	#72280647, #72281946, #72282134, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401956);