#138355

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

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

normal subgroups only

This group has 541 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:

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

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,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, 45551);