#147841

order	512
nilpotency class	4
isoclinism	isoclinic class of #13353
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2021
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

1032 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:

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

Polycyclic presentation:

2^9|a^2=g,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=eh,f^b=fi,g^b=gh,d^c=dh,f^c=fi,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, 55037);