#149275

order	512
nilpotency class	4
isoclinism	isoclinic class of #13350
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2021
commutator	C₂×#41	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂	#11344
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

670 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56471);