#136118

order	512
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1852
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2922 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

242 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 80 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)(33,34,39,45)(35,47,36,50)(37,53,56,62)(38,46,40,44)(41,51,60,48)(42,49,61,52)(43,55,58,57)(54,63,64,59);(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)(33,35,39,36)(34,41,45,60)(37,54,56,64)(38,49,40,52)(42,58,61,43)(44,59,46,63)(47,62,50,53)(48,55,51,57);(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)(33,36,39,35)(34,42,45,61)(37,51,56,48)(38,49,40,52)(41,58,60,43)(44,47,46,50)(53,59,62,63)(54,57,64,55)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=gi,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,g^b=gi,d^c=dh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43314);