#148352

order	512
nilpotency class	4
isoclinism	isoclinic class of #135851
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1854
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

641 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,f^b=fi,d^c=di,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, 55548);