#150245

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	C₂²×C₁₆	C₂³
prime radical	C₂⁴	D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 313 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=gh,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, 57441);