#150945

order	512
nilpotency class	5
isoclinism	isoclinic class of #150243
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	C₂²×D₁₆	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

288 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=gh,e^2=fghi,f^2=hi,g^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gh,h^a=hi,c^b=ci,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58141);