#136048

order	512
nilpotency class	4
isoclinism	isoclinic class of #12170
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1598
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8856
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

325 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 325 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=h,d^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dg,e^b=ehi,g^b=gi,d^c=dh,e^c=ei,e^d=ei

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, 43244);