#123049

order	512
nilpotency class	4
isoclinism	isoclinic class of #123031
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

276 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 42 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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,28)(14,27)(15,26)(16,22)(17,23)(18,25)(19,29)(20,30)(21,31)(24,32)(33,34,39,45)(35,46,47,61)(36,40,51,56)(37,44,38,43)(41,52,59,49)(42,53,55,50)(48,60,62,58)(54,64,57,63);(1,3,6,14)(2,7,10,20)(4,15,17,12)(5,13,18,30)(8,21,23,19)(9,11,25,27)(16,31,32,29)(22,26,24,28)(34,40)(35,47)(36,50)(37,38)(41,57)(42,44)(43,55)(45,56)(46,58)(48,63)(51,53)(54,59)(60,61)(62,64);(1,4,6,17)(2,8,10,23)(3,12,14,15)(5,16,18,32)(7,19,20,21)(9,24,25,22)(11,26,27,28)(13,29,30,31)(33,35,39,47)(34,41,45,59)(36,48,51,62)(37,52,38,49)(40,54,56,57)(42,60,55,58)(43,61,44,46)(50,64,53,63)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,d^2=fh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^c=eh,e^d=ei

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30245);