#512831

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
prime radical	C₂²×#508	C₂
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

763 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 763 conjugacy classes of subgroups — too many to draw. Only its 148 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)(5,6,11,16)(7,17,22,48)(8,23,26,13)(9,27,30,61)(10,31,33,55)(12,34,38,19)(14,40,42,29)(15,43,45,25)(18,47,21,46)(20,52,53,66)(24,56,59,60)(28,44,63,64)(32,41,50,68)(35,49,37,51)(36,57,65,62)(39,58,54,67);(5,7)(6,12)(8,18)(9,28)(10,32)(11,22)(13,35)(14,41)(15,44)(16,38)(17,27)(19,29)(20,25)(21,26)(23,37)(24,57)(30,63)(31,36)(33,50)(34,40)(39,66)(42,68)(43,53)(45,64)(46,60)(47,56)(48,61)(49,58)(51,67)(52,54)(55,65)(59,62);(2,4);(5,8,11,26)(6,13,16,23)(7,18,22,21)(9,24,30,59)(10,25,33,43)(12,35,38,37)(14,39,42,54)(15,31,45,55)(17,46,48,47)(19,49,34,51)(20,50,53,32)(27,60,61,56)(28,57,63,62)(29,58,40,67)(36,64,65,44)(41,66,68,52)

Polycyclic presentation:

2^9|a^2=i,d^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, 420027);