#513182

order	512
nilpotency class	5
isoclinism	isoclinic class of #513173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

452 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 120 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:

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

Polycyclic presentation:

2^9|a^2=i,e^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dghi,e^a=eg,g^a=gh,h^a=hi,c^b=ci,d^b=de,e^b=eg,g^b=gh,h^b=hi,e^d=eg,g^d=gh,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420378);