#149363

order	512
nilpotency class	4
isoclinism	isoclinic class of #13545
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2101
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11372
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

427 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,d^2=h,e^2=h,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=dh,e^b=eh,g^b=gh,h^b=hi,d^c=dhi,e^c=eh,g^c=ghi,h^c=hi

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, 56559);