#150744

order	512
nilpotency class	5
isoclinism	isoclinic class of #150413
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	#11840
prime radical	#2017	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2922 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

241 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 241 conjugacy classes of subgroups — too many to draw. Only its 50 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=h,d^2=gi,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=cfi,d^b=dg,e^b=efg,g^b=gi,h^b=hi,d^c=dh,f^c=fi,e^d=eh,f^d=fi,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57940);