#153407

order	512
nilpotency class	5
isoclinism	isoclinic class of #153255
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2233
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	C₂×SD₃₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

211 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 211 conjugacy classes of subgroups — too many to draw. Only its 72 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 — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=dh,b^2=deg,c^2=i,d^2=ghi,e^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,d^b=dg,e^b=eg,g^b=gh,h^b=hi,d^c=dgi,e^c=egi,g^c=gh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60603);