#151032

order	512
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	D₁₆
commutator	C₁₆	C₂²×C₈
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂²	C₂×#366
prime radical	C₂³	#366
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

152 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=df,c^2=i,d^2=fh,e^2=g,f^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58228);