#500941

order	512
nilpotency class	4
isoclinism	isoclinic class of #500940
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24338
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2961 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

679 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=ei,f^2=h,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=gh,h^b=hi,d^c=dh,e^c=ei,f^c=fhi,g^c=ghi,h^c=hi,g^d=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307107, #72307114, #72307115, …
maximal normal subgroup of	#72307107, #72307114, #72307115, …
maximal quotient of	C₂×#500941

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408137);