#151737

order	512
nilpotency class	5
isoclinism	isoclinic class of #151734
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²	#2246
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

186 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58933);