#506267

order	512
nilpotency class	4
isoclinism	isoclinic class of #506267
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#28143
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

558 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 128 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=g,d^2=eh,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^c=ei,g^c=gh,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413463);