#150056

order	512
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

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×#357
prime radical	C₂×#508	C₄
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

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

normal subgroups only

This group has 264 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)(12,17)(13,31)(14,35)(15,21)(16,24)(18,33)(19,29)(20,30)(23,49)(25,55)(26,57)(27,50)(28,58)(32,46)(34,64)(36,43)(37,42)(38,51)(39,65)(40,54)(41,60)(44,70)(45,67)(47,62)(48,53)(52,69)(56,71)(59,68)(61,63)(66,72);(9,11)(10,16)(12,23)(13,32)(14,36)(15,27)(17,38)(18,43)(19,46)(20,48)(21,40)(22,31)(24,33)(25,29)(26,30)(28,59)(34,66)(35,39)(37,52)(41,55)(42,68)(44,71)(45,69)(47,72)(49,58)(50,64)(51,60)(53,70)(54,67)(56,62)(57,63)(61,65);(9,12,15,30)(10,17,21,20)(11,23,27,26)(13,28,34,61)(14,29,37,62)(16,38,40,48)(18,41,45,44)(19,42,47,35)(22,49,50,57)(24,51,54,53)(25,52,56,36)(31,58,64,63)(32,59,66,65)(33,60,67,70)(39,46,68,72)(43,55,69,71)

Polycyclic presentation:

2^9|a^2=e,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,d^b=df,f^b=fh,h^b=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, 57252);