#497409

order	512
nilpotency class	4
isoclinism	isoclinic class of #497408
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₄×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61105	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1042 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)(3,17)(4,23,27,58)(5,12)(6,15)(7,32)(8,13)(9,16)(10,20)(11,26,41,50)(14,19)(18,39,53,64)(21,44)(22,45)(24,36,55,61)(25,47,35,57)(28,46)(29,38)(30,43)(31,42)(33,48,60,40)(34,37)(49,52,62,59)(51,56,54,63);(1,3,9,22)(2,10,16,38)(4,18,27,53)(5,7,21,34)(6,29,8,20)(11,36,41,61)(12,42,44,14)(13,45,15,17)(19,30,31,28)(23,56,58,63)(24,33,55,60)(25,54,35,51)(26,52,50,59)(32,43,37,46)(39,57,64,47)(40,62,48,49);(1,4)(2,11)(3,18)(5,24)(6,25)(7,33)(8,35)(9,27)(10,36)(12,39)(13,40)(14,47)(15,48)(16,41)(17,49)(19,50)(20,51)(21,55)(22,53)(23,37)(26,31)(28,52)(29,54)(30,59)(32,58)(34,60)(38,61)(42,57)(43,63)(44,64)(45,62)(46,56);(1,5)(2,12)(3,19)(4,24)(6,30)(7,29)(8,28)(9,21)(10,37)(11,39)(13,46)(14,17)(15,43)(16,44)(18,50)(20,34)(22,31)(23,36)(25,59)(26,53)(27,55)(32,38)(33,54)(35,52)(40,56)(41,64)(42,45)(47,49)(48,63)(51,60)(57,62)(58,61)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285521, #72285570, #72285581
maximal normal subgroup of	#72285521, #72285570, #72285581
maximal quotient of	#72287396, #72287467, #72287554, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404605);