#148888

order	512
nilpotency class	4
isoclinism	isoclinic class of #13469
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2065
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11231
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

459 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)(3,16,20,35)(4,21)(5,27)(6,30)(7,14)(8,13)(9,15)(10,18,37,31)(11,24)(12,29)(17,38,48,63)(19,51,32,45)(22,34,49,62)(23,40)(25,59)(26,54)(28,41)(33,53)(36,55,42,50)(39,46)(43,58)(44,60,61,47)(52,56,64,57);(1,3,5,18,8,19,29,50,9,20,6,31,7,32,28,55)(2,10,12,16,14,36,27,45,15,37,41,35,13,42,30,51)(4,22,23,56,25,17,46,47,26,49,24,57,33,48,58,60)(11,38,21,52,39,34,53,44,40,63,54,64,43,62,59,61);(1,4,5,23,8,25,29,46,9,26,6,24,7,33,28,58)(2,11,12,21,14,39,27,53,15,40,41,54,13,43,30,59)(3,17,18,47,19,49,50,57,20,48,31,60,32,22,55,56)(10,34,16,44,36,63,45,64,37,62,35,61,42,38,51,52)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,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, 56084);