#150982

order	512
nilpotency class	5
isoclinism	isoclinic class of #150785
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

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	#33713	C₂
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

275 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 275 conjugacy classes of subgroups — too many to draw. Only its 76 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)(9,10,14,21,32,53,68,51,17,19,13,33,16,23,39,55,15,22,37,54,35,48,36,70,12,30,31,20,40,50,63,72)(11,24,27,58,64,52,67,18,25,44,26,61,29,49,65,47,28,59,60,46,41,71,34,45,42,57,62,56,66,69,38,43);(2,5)(3,7)(9,11)(10,18)(12,25)(13,34)(14,38)(15,28)(16,41)(17,42)(19,43)(20,49)(21,52)(22,45)(23,56)(24,51)(26,36)(27,63)(29,35)(30,47)(31,67)(32,66)(33,69)(37,65)(39,60)(40,64)(44,72)(46,48)(50,61)(53,58)(54,71)(55,57)(59,70)(62,68);(2,5)(6,8)(9,12,15,17)(10,19,22,30)(11,25,28,42)(13,14,31,37)(16,32,40,35)(18,43,45,47)(20,21,33,54)(23,48,50,53)(24,57,59,44)(26,60,62,27)(29,41,66,64)(34,38,67,65)(36,39,68,63)(46,61,58,56)(49,52,69,71)(51,55,70,72)

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=gh,e^2=fgh,g^2=hi,h^2=i,b^a=bd,c^a=cf,d^a=df,d^b=dg,e^b=efg,g^b=gh,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58178);