#135701

order	512
nilpotency class	4
isoclinism	isoclinic class of #135685
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1834
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#487
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

181 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 181 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=dg,d^2=hi,e^2=gi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ci,f^b=fghi,f^c=fghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42897);