#150943

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	C₄×C₁₆	C₂³
socle	C₂²	C₂×#357
prime radical	#33713	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

286 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

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,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, 58139);