#150475

order	512
nilpotency class	5
isoclinism	isoclinic class of #150474
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11773
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

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

normal subgroups only

This group has 262 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)(4,10,23,39)(5,26,25,40)(7,14,32,47)(8,13,31,44)(9,16,35,27)(11,42,45,29)(17,48,54,46)(19,43,30,61)(20,51,52,62)(21,56,57,36)(22,38,58,64)(24,41,28,49)(33,59,60,34)(37,53,63,55);(1,3)(2,9,13,36)(4,21)(5,27)(6,30)(7,33)(8,20)(10,34,41,61)(11,43,26,59)(12,46,44,63)(14,48,38,37)(15,49,51,39)(16,24)(17,28)(18,32)(19,22)(23,55)(25,53)(29,57)(31,60)(35,47,56,64)(40,62,45,50)(42,54)(52,58);(1,4,22,42,8,24,7,25)(2,10,38,45,13,41,14,40)(3,16,33,54,20,21,19,53)(5,6,23,58,29,31,28,32)(9,34,48,62,36,61,37,50)(11,44,49,47,26,12,39,64)(15,35,59,46,51,56,43,63)(17,52,57,30,55,18,27,60)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=gi,d^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cf,d^b=dg,e^b=efgi,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, 57671);