#141371

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	#19667	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 298 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)(35,43)(36,40,50,54)(37,46)(38,42)(39,45)(41,52)(44,58,62,56)(47,57)(48,59,63,49)(51,61,64,55)(53,60);(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)(33,35,43,34,39,46,53,41,38,47,57,42,52,60,37,45)(36,48,58,61,51,62,49,40,50,63,56,55,64,44,59,54);(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)(33,36,39,51,38,50,52,64)(34,40,45,61,42,54,41,55)(35,44,60,63,47,62,46,48)(37,49,43,56,53,59,57,58)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=egi,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, 48567);