#150927

order	512
nilpotency class	5
isoclinism	isoclinic class of #150924
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11903
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	#423	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

237 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gi,d^2=ghi,e^2=hi,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,c^b=cf,d^b=dg,e^b=efgh,h^b=hi,d^c=dhi,e^d=eh,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58123);