#138267

order	512
nilpotency class	4
isoclinism	isoclinic class of #138264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9147
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

719 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 719 conjugacy classes of subgroups — too many to draw. Only its 100 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,d^2=gh,e^2=i,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,d^c=di,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45463);