#150902

order	512
nilpotency class	5
isoclinism	isoclinic class of #150414
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2128
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	#408	C₂³
socle	C₂	#11899
prime radical	#532	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 241 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fh,c^2=gh,d^2=gh,e^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=cf,d^b=dg,e^b=efghi,f^b=fi,h^b=hi,d^c=dhi,e^c=ei,f^c=fi,e^d=eh,f^d=fi,f^e=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, 58098);