#150900

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

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

normal subgroups only

This group has 243 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)(3,14,17,45)(4,20,21,54)(5,23,25,41)(7,13,30,42)(8,12,29,43)(9,32,34,24)(10,37,39,26)(15,36,48,61)(16,50,51,40)(18,33,53,60)(19,46,52,64)(22,44,28,59)(27,57,47,56)(31,38,55,63)(35,58,62,49);(1,3,7,19)(2,9,13,35)(4,16)(5,24)(6,27)(8,18,31,47)(10,33)(11,40)(12,36,38,50)(14,44)(15,22)(17,29)(20,46)(21,32,28,58)(23,56)(25,48,37,51)(26,49)(30,53)(34,43)(39,64,41,45)(42,61)(52,55)(54,57,59,60)(62,63);(1,4,8,5,7,22,31,26)(2,10,38,44,13,23,12,20)(3,15,47,24,19,16,18,49)(6,28,29,37,30,21,55,25)(9,33,36,46,35,56,50,14)(11,41,63,54,42,39,43,59)(17,51,27,58,52,48,53,32)(34,57,61,45,62,60,40,64)

Polycyclic presentation:

2^9|a^2=e,b^2=fh,c^2=gh,d^2=gh,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, 58096);