#492930

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1746
prime radical	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

597 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,f^2=h,h^2=i,b^a=be,d^a=di,g^a=gi,c^b=ch,e^b=ei,g^c=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239196, #72239198, #72239200
maximal normal subgroup of	#72239196, #72239198, #72239200

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400126);