#141370

order	512
nilpotency class	4
isoclinism	isoclinic class of #141367
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1961
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

281 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 281 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eg,d^2=ghi,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,f^c=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48566);