#511895

order	512
nilpotency class	4
isoclinism	isoclinic class of #511893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1053 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 1053 conjugacy classes of subgroups — too many to draw. Only its 180 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(2,4)(5,6,12,18)(7,19,24,37)(8,25,28,58)(9,17,11,15)(10,30,32,42)(13,23,38,21)(14,39,41,27)(16,29,44,33)(20,48,51,68)(22,36,54,64)(26,59,34,43)(31,45,61,40)(35,62,63,53)(46,66,65,50)(47,52,60,55)(49,57,56,67);(1,2,3,4)(5,7,11,23,12,24,9,21)(6,13,17,37,18,38,15,19)(8,20,34,56,28,51,26,49)(10,22,33,55,32,54,29,52)(14,35,45,65,41,63,40,46)(16,36,30,60,44,64,42,47)(25,57,59,68,58,67,43,48)(27,50,61,62,39,66,31,53);(5,8)(6,14)(7,20)(9,26)(10,31)(11,34)(12,28)(13,35)(15,40)(16,43)(17,45)(18,41)(19,46)(21,49)(22,53)(23,56)(24,51)(25,30)(27,33)(29,39)(32,61)(36,48)(37,65)(38,63)(42,58)(44,59)(47,67)(50,55)(52,66)(54,62)(57,60)(64,68);(10,32)(16,44)(22,54)(25,58)(27,39)(29,33)(30,42)(31,61)(36,64)(43,59)(47,60)(48,68)(50,66)(52,55)(53,62)(57,67)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fh,h^a=hi,f^c=fh,g^c=gi,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, 419091);