#123048

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,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, 30244);