#146704

order	512
nilpotency class	4
isoclinism	isoclinic class of #12707
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1781
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10761
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,d^b=di,f^b=fhi,e^c=ei,f^c=fhi,f^d=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312707, #72313027, #72313033, …
maximal normal subgroup of	#72312707, #72313027, #72313033, …
maximal quotient of	C₂×#146704

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53900);