#493915

order	512
nilpotency class	4
isoclinism	isoclinic class of #12873
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1823
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1823
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

997 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)(3,14,18,9)(4,20,23,55)(5,11)(6,25)(7,29)(8,12)(10,22,35,56)(13,27)(15,42)(16,40,19,32)(17,47,49,57)(21,54,30,38)(24,37)(26,36,58,34)(28,53)(31,44)(33,50,41,46)(39,62)(43,61)(45,52)(48,59)(51,63)(60,64);(1,3)(2,9)(4,15)(5,16)(6,17)(7,19)(8,18)(10,31)(11,32)(12,14)(13,33)(20,52)(21,43)(22,44)(23,45)(24,46)(25,57)(26,48)(27,50)(28,49)(29,40)(30,51)(34,59)(35,39)(36,60)(37,41)(38,61)(42,55)(47,53)(54,63)(56,62)(58,64);(1,4)(2,10,12,35)(3,15)(5,21)(6,26)(7,30)(8,23)(9,31,14,39)(11,34,29,36)(13,38,37,54)(16,43)(17,48)(18,45)(19,51)(20,53,55,25)(22,27)(24,56)(28,58)(32,59,40,60)(33,61,41,63)(42,57,52,47)(44,50)(46,62)(49,64);(1,5,8,7)(2,11,12,29)(3,16,18,19)(4,21,23,30)(6,24,28,27)(9,32,14,40)(10,34,35,36)(13,25,37,53)(15,43,45,51)(17,46,49,50)(20,54,55,38)(22,26,56,58)(31,59,39,60)(33,57,41,47)(42,61,52,63)(44,48,62,64)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307096, #72307101, #72307111, …
maximal normal subgroup of	#72307096, #72307101, #72307111, …
maximal quotient of	C₂×#493915

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401111);