#148346

order	512
nilpotency class	4
isoclinism	isoclinic class of #13318
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1999
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

647 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:

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

Polycyclic presentation:

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

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, 55542);