#150248

order	512
nilpotency class	5
isoclinism	isoclinic class of #7418
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×#357
prime radical	#33600	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 310 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)(9,10,12,18,16,23,34,22)(11,24,27,59,30,61,25,58)(13,35,31,67,39,53,70,71)(14,40,32,20,15,42,33,21)(17,44,46,26,49,62,45,64)(19,51,50,38,55,69,54,37)(28,65,41,60,29,66,43,52)(36,56,72,47,63,57,68,48);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,25)(13,36)(14,41)(15,43)(16,30)(18,45)(19,52)(20,56)(21,57)(22,46)(23,49)(24,53)(26,38)(27,34)(28,32)(29,33)(31,68)(35,61)(37,64)(39,63)(40,47)(42,48)(44,69)(50,66)(51,62)(54,65)(55,60)(58,71)(59,67)(70,72);(9,12,16,34)(10,18,23,22)(11,25,30,27)(13,31,39,70)(14,32,15,33)(17,45,49,46)(19,50,55,54)(20,42,21,40)(24,58,61,59)(26,44,64,62)(28,43,29,41)(35,67,53,71)(36,68,63,72)(37,51,38,69)(47,56,48,57)(52,66,60,65)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57444);