#151128

order	512
nilpotency class	5
isoclinism	isoclinic class of #151127
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²	#2177
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=eh,d^2=fhi,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=cgh,d^a=df,f^a=fh,h^a=hi,c^b=cg,d^b=dfi,e^b=ei,f^b=fh,h^b=hi,d^c=di

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232016, #72232023, #72232025, …
maximal normal subgroup of	#72232016, #72232023, #72232025, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58324);