#150345

order	512
nilpotency class	5
isoclinism	isoclinic class of #150171
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#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 ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2929 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 330 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)(3,15,18,52)(4,10,22,39)(5,11,27,41)(7,32,31,62)(8,13,30,48)(9,21,35,25)(14,49,47,26)(16,51,54,36)(17,37,55,45)(19,60,59,34)(20,53,58,63)(23,46,28,44)(24,40,61,43)(29,38,33,42)(50,57,64,56);(1,3)(2,9)(4,21)(5,25)(6,29)(7,33)(8,20)(10,38)(11,42)(12,45)(13,36)(14,50)(15,43)(16,24)(17,22)(18,49)(19,26)(23,55)(27,57)(28,54)(30,59)(31,58)(32,37)(34,40)(35,47)(39,53)(41,52)(44,60)(46,63)(48,64)(51,62)(56,61);(1,4,7,23,8,24,26,27)(2,10,32,46,13,40,14,41)(3,16,33,57,20,21,19,55)(5,6,22,31,28,30,61,49)(9,34,37,52,36,38,50,63)(11,12,39,62,44,48,43,47)(15,51,42,64,53,35,60,45)(17,18,54,29,56,58,25,59)

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,c^b=ci,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57541);