#138352

order	512
nilpotency class	4
isoclinism	isoclinic class of #138352
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9155
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

533 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 533 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45548);