#153103

order	512
nilpotency class	5
isoclinism	isoclinic class of #153078
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

383 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,4)(9,10,15,21)(11,22,26,55)(12,17,31,44)(13,32,35,67)(14,36,34,20)(16,38,41,23)(18,45,48,28)(19,49,51,29)(24,57,58,71)(25,59,37,54)(27,50,62,72)(30,53,63,43)(33,46,68,70)(39,65,69,61)(40,56,52,60)(42,66,47,64);(2,4)(5,6)(7,8)(9,11,15,26)(10,16,21,41)(12,27,31,62)(13,33,35,68)(14,37,34,25)(17,22,44,55)(18,46,48,70)(19,50,51,72)(20,52,36,40)(23,45,38,28)(24,49,58,29)(30,65,63,61)(32,39,67,69)(42,57,47,71)(43,59,53,54)(56,64,60,66);(6,8)(10,17)(12,28)(13,29)(14,34)(16,22)(18,19)(20,53)(21,44)(23,27)(24,33)(25,37)(30,66)(31,45)(32,42)(35,49)(36,43)(38,62)(39,57)(40,59)(41,55)(46,50)(47,67)(48,51)(52,54)(56,65)(58,68)(60,61)(63,64)(69,71)(70,72)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gh,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=eg,g^b=gh,h^b=hi,d^c=dg,e^c=eg,g^c=gh,h^c=hi

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, 60299);