#500169

order	512
nilpotency class	4
isoclinism	isoclinic class of #500168
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	#61081	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2932 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

936 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)(3,15)(4,21)(5,6)(7,13)(8,29)(9,31)(10,17)(11,25)(12,33)(14,23)(16,51)(18,39)(19,45)(20,40)(22,59)(24,60)(26,43)(27,44)(28,46)(30,32)(34,38)(35,47)(36,42)(37,57)(41,58)(48,62)(49,52)(50,55)(53,61)(54,63)(56,64);(1,3)(2,10)(4,22)(5,15)(6,17)(7,34)(8,18)(9,35)(11,41)(12,39)(13,45)(14,47)(16,26)(19,32)(20,23)(21,59)(24,61)(25,58)(27,52)(28,56)(29,50)(30,38)(31,40)(33,55)(36,57)(37,42)(43,51)(44,49)(46,64)(48,63)(53,60)(54,62);(1,4,23,27,6,28,9,36)(2,11,31,43,5,24,14,48)(3,16,35,53,17,54,20,58)(7,26,8,25,32,62,33,60)(10,37,47,59,15,49,40,64)(12,44,30,21,29,42,13,46)(18,56,34,52,55,22,19,57)(38,41,39,63,45,61,50,51);(1,5,6,2)(3,15,17,10)(4,24,28,11)(7,12,32,29)(8,30,33,13)(9,31,23,14)(16,49,54,37)(18,38,55,45)(19,50,34,39)(20,47,35,40)(21,60,46,25)(22,61,56,41)(26,44,62,42)(27,48,36,43)(51,52,63,57)(53,64,58,59)

Polycyclic presentation:

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

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307109, #72307114, #72307115, …
maximal normal subgroup of	#72307109, #72307114, #72307115, …
maximal quotient of	C₂×#500169

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407365);