#150941

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	C₂³	#357
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

278 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 94 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)(11,22)(12,17)(13,31)(14,35)(15,21)(16,24)(18,33)(19,29)(20,30)(23,50)(25,56)(26,52)(27,51)(28,61)(32,47)(34,49)(36,44)(37,43)(38,53)(39,66)(40,59)(41,55)(42,63)(45,72)(46,54)(48,57)(58,70)(60,71)(62,69)(64,65)(67,68);(2,5)(3,7)(9,11,15,26)(10,16,21,41)(12,27,30,23)(13,32,34,67)(14,36,37,60)(17,40,20,38)(18,44,46,71)(19,47,48,68)(22,49,52,31)(24,54,55,33)(25,57,58,29)(28,62,64,66)(35,39,43,69)(42,56,45,70)(50,61,51,65)(53,63,59,72);(9,12,15,30)(10,17,21,20)(11,23,26,27)(13,28,34,64)(14,29,37,57)(16,38,41,40)(18,42,46,45)(19,43,48,35)(22,50,52,51)(24,53,55,59)(25,36,58,60)(31,61,49,65)(32,66,67,62)(33,63,54,72)(39,68,69,47)(44,70,71,56)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=gh,e^2=fgh,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, 58137);