#150344

order	512
nilpotency class	5
isoclinism	isoclinic class of #7396
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#353
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	#408	C₂³
socle	C₂	#11726
prime radical	#23794	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2929 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 333 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)(3,15,18,49)(4,10,21,37)(5,11,26,39)(7,31,30,60)(8,13,29,46)(9,16,34,24)(14,47,45,25)(17,36,54,43)(19,57,56,61)(20,50,51,62)(22,44,27,42)(23,38,58,41)(28,33,32,40)(35,53,63,59)(48,52,64,55);(1,3)(2,9)(4,16)(5,24)(6,28)(7,32)(8,20)(10,33)(11,40)(12,43)(13,35)(14,48)(15,41)(17,21)(18,47)(19,25)(22,54)(23,53)(26,52)(27,59)(29,56)(30,51)(31,36)(34,45)(37,50)(38,61)(39,49)(42,57)(44,62)(46,64)(55,58)(60,63);(1,4,7,22,8,23,25,26)(2,10,31,44,13,38,14,39)(3,16,32,54,20,53,19,52)(5,6,21,30,27,29,58,47)(9,33,36,62,35,61,48,49)(11,12,37,60,42,46,41,45)(15,34,40,43,50,63,57,64)(17,51,59,56,55,18,24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57540);