#142397

order	512
nilpotency class	4
isoclinism	isoclinic class of #142396
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9920
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

450 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 68 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)(3,13,18,44)(4,19,23,38)(5,24,28,59)(6,29,26,11)(8,30,33,15)(9,22,37,21)(10,39,41,27)(14,45,48,63)(16,43,17,40)(20,36,56,64)(25,34,50,32)(31,60,35,54)(42,51,61,47)(46,52,49,62)(53,55,58,57);(1,3,6,16,7,18,26,17)(2,8,11,32,12,33,29,34)(4,14,22,51,23,48,21,47)(5,25,27,30,28,50,39,15)(9,35,38,62,37,31,19,52)(10,40,24,44,41,43,59,13)(20,54,57,49,56,60,55,46)(36,63,58,61,64,45,53,42);(1,4)(2,9)(3,14)(5,20)(6,22)(7,23)(8,31)(10,36)(11,38)(12,37)(13,42)(15,49)(16,51)(17,47)(18,48)(19,29)(21,26)(24,58)(25,60)(27,57)(28,56)(30,46)(32,52)(33,35)(34,62)(39,55)(40,63)(41,64)(43,45)(44,61)(50,54)(53,59)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49593);