#150465

order	512
nilpotency class	5
isoclinism	isoclinic class of #150463
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11772
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	#408	C₂³
prime radical	#24279	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2929 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

420 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 54 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)(3,15,18,48)(4,10,24,11)(5,27,25,41)(7,14,23,39)(8,13,31,46)(9,33,35,58)(16,45,17,37)(19,42,54,64)(20,49,53,34)(21,55,51,47)(22,38,52,43)(26,40,57,44)(28,36,50,62)(29,60,56,63)(30,61,32,59);(1,3)(2,9)(4,21)(5,28)(6,30)(7,32)(8,20)(10,34)(11,42)(12,45)(13,36)(14,47)(15,44)(16,25)(17,29)(18,52)(19,22)(23,53)(24,58)(26,50)(27,59)(31,54)(33,56)(35,39)(37,38)(40,64)(41,48)(43,62)(46,55)(49,60)(51,57)(61,63);(1,4,22,56,8,25,7,26)(2,10,38,63,13,41,14,40)(3,16,32,33,20,21,19,50)(5,23,57,6,24,52,29,31)(9,34,47,64,36,48,37,61)(11,43,60,46,27,39,44,12)(15,45,59,35,49,55,42,62)(17,30,58,53,51,54,28,18)

Polycyclic presentation:

2^9|a^2=e,c^2=gi,d^2=gi,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cf,d^b=dg,e^b=efg,g^b=gi,d^c=dh,f^c=fi,e^d=eh,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, 57661);