#149047

order	512
nilpotency class	4
isoclinism	isoclinic class of #140091
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1918
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11292
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

449 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=h,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,d^b=dh,f^b=fhi,g^b=ghi,h^b=hi,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, 56243);