#148347

order	512
nilpotency class	4
isoclinism	isoclinic class of #13497
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2075
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11096
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

653 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)(3,14)(4,21,25,10)(5,26)(6,13)(7,31)(8,12)(9,17)(11,15)(16,41,48,62)(18,44)(19,43)(20,53)(22,55,52,64)(23,58,27,60)(24,38,30,37)(28,40)(29,59)(32,47,63,49)(33,51)(34,50)(35,57,54,61)(36,56,39,46)(42,45);(1,3,15,45,7,20,29,50,8,18,28,51,6,19,5,17)(2,9,26,43,13,33,40,44,12,34,59,53,31,42,11,14)(4,22,56,49,24,48,27,61,25,52,46,47,30,16,23,57)(10,35,60,62,38,32,39,55,21,54,58,41,37,63,36,64);(1,4)(2,10)(3,16)(5,27)(6,30)(7,24)(8,25)(9,32)(11,39)(12,21)(13,38)(14,41)(15,46)(17,49)(18,48)(19,52)(20,22)(23,29)(26,58)(28,56)(31,37)(33,54)(34,63)(35,42)(36,40)(43,64)(44,62)(45,61)(47,50)(51,57)(53,55)(59,60)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55543);