#512829

order	512
nilpotency class	5
isoclinism	isoclinic class of #512827
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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 ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

765 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 765 conjugacy classes of subgroups — too many to draw. Only its 146 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)(7,18)(8,13)(9,27)(10,31)(11,14)(12,20)(15,29)(16,25)(17,26)(19,45)(21,51)(22,53)(23,46)(24,54)(28,43)(30,59)(32,41)(33,38)(34,47)(35,50)(36,61)(37,56)(39,63)(40,58)(42,67)(44,49)(48,65)(52,68)(55,64)(57,60)(62,66);(5,7)(6,12)(8,19)(9,28)(10,32)(11,23)(13,34)(14,35)(15,41)(16,43)(17,44)(18,27)(20,29)(21,25)(22,26)(24,55)(30,62)(31,36)(33,48)(37,51)(38,64)(39,65)(40,66)(42,68)(45,54)(46,59)(47,56)(49,67)(50,63)(52,58)(53,60)(57,61);(2,4)(5,8,11,26)(6,13,14,17)(7,19,23,22)(9,24,30,57)(10,25,33,58)(12,34,35,44)(15,37,39,42)(16,38,40,31)(18,45,46,53)(20,47,50,49)(21,48,52,32)(27,54,59,60)(28,55,62,61)(29,56,63,67)(36,43,64,66)(41,51,65,68);(6,14)(12,35)(13,17)(15,39)(16,40)(18,46)(27,59)(31,38)(34,44)(36,64)(37,42)(41,65)(43,66)(45,53)(51,68)(54,60)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420025);