#136123

order	512
nilpotency class	4
isoclinism	isoclinic class of #136011
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1853
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2922 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

240 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,d^2=g,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dg,e^b=eh,f^b=fi,g^b=gi,d^c=dh,e^c=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, 43319);