#150464

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	#525	D₄
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

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

normal subgroups only

This group has 415 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)(3,15,18,50)(4,10,22,38)(5,25,28,46)(7,14,32,47)(8,13,31,42)(9,21,35,58)(11,24,39,29)(16,49,26,36)(17,37,54,45)(19,41,57,61)(20,51,56,63)(23,40,53,44)(27,43,60,62)(30,34,33,59)(48,52,64,55);(1,3,8,20)(2,9,13,36)(4,21,23,16)(5,26,29,58)(6,30,31,57)(7,33,27,19)(10,34,40,61)(11,41,25,59)(12,45,42,64)(14,48,43,37)(15,38,51,44)(17,53,55,22)(18,32,56,60)(24,52,28,54)(35,62,49,47)(39,63,46,50);(1,4,7,24,8,23,27,28)(2,10,14,39,13,40,43,46)(3,16,19,52,20,21,33,54)(5,6,22,32,29,31,53,60)(9,34,37,50,36,61,48,63)(11,42,44,62,25,12,38,47)(15,49,41,64,51,35,59,45)(17,18,26,57,55,56,58,30)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=g,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, 57660);