#147854

order	512
nilpotency class	4
isoclinism	isoclinic class of #13432
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2044
commutator	C₂×#41	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10995
prime radical	#60749	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1036 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)(3,13)(4,18,22,9)(5,23,24,10)(6,12)(7,11)(8,15,30,46)(14,40,44,58)(16,42)(17,41)(19,49,47,60)(20,33)(21,39,27,34)(25,53,54,36)(26,57)(28,37)(29,43)(31,45,38,55)(32,50)(35,51)(48,64)(52,61)(56,62)(59,63);(1,3)(2,8)(4,19)(5,15)(6,26)(7,16)(9,32)(10,13)(11,30)(12,38)(14,21)(17,28)(18,48)(20,50)(22,47)(23,42)(24,46)(25,55)(27,44)(29,34)(31,37)(33,60)(35,49)(36,57)(39,59)(40,62)(41,53)(43,52)(45,54)(51,64)(56,63)(58,61);(1,4)(2,9)(3,14)(5,20)(6,27)(7,22)(8,29)(10,35)(11,18)(12,39)(13,40)(15,43)(16,44)(17,47)(19,26)(21,28)(23,33)(24,51)(25,56)(30,59)(31,48)(32,38)(34,37)(36,61)(41,60)(42,58)(45,64)(46,63)(49,57)(50,55)(52,54)(53,62)

Polycyclic presentation:

2^9|a^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,e^b=ehi,f^b=fi,g^b=gh,d^c=dh,f^c=fi,g^c=gi,e^d=ei,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55050);