#513053

order	512
nilpotency class	5
isoclinism	isoclinic class of #512939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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 ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

548 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 162 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)(7,17)(8,23)(9,13)(10,29)(11,30)(12,20)(14,18)(15,24)(16,28)(19,37)(21,50)(22,51)(25,52)(26,56)(27,57)(31,45)(32,60)(33,61)(34,42)(35,55)(36,62)(38,41)(39,48)(40,47)(43,54)(44,53)(46,67)(49,58)(59,68)(63,65)(64,66);(5,7,18,41,10,21,43,61,28,47,62,68,55,66,25,45,9,19,42,65,27,46,26,49,11,22,44,60,24,48,8,20)(6,12,23,39,15,32,53,51,30,58,56,67,57,63,34,37,13,31,52,64,35,59,36,40,16,33,54,50,29,38,14,17);(1,2,3,4)(6,13)(12,31)(14,34)(15,35)(16,30)(17,37)(23,52)(29,57)(32,59)(33,58)(36,53)(38,63)(39,64)(40,51)(50,67)(54,56);(7,19)(12,31)(17,37)(20,45)(21,46)(22,47)(32,59)(33,58)(38,63)(39,64)(40,51)(41,65)(48,66)(49,61)(50,67)(60,68)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420249);