#123208

order	512
nilpotency class	4
isoclinism	isoclinic class of #123191
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator, Frattini, cosocle	C₂×C₄⋊C₈	C₂³
prime radical	C₂³	#447
LCS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 226 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)(35,39)(36,46)(37,50)(38,42)(40,49)(41,51)(43,58)(44,57)(45,54)(47,62)(48,61)(52,59)(53,60)(55,63)(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)(33,35,38,45)(34,39,42,54)(36,47,48,43)(37,44,51,60)(40,55,56,52)(41,57,50,53)(46,58,61,62)(49,63,64,59);(1,3,7,17)(2,9,13,27)(4,19,21,15)(5,22,6,16)(8,23,25,30)(10,28,24,26)(11,29,12,14)(18,32,20,31)(33,36)(34,40,42,56)(35,43)(37,49)(38,48)(39,52,54,55)(41,46,50,61)(44,59)(45,47)(51,64)(53,58,57,62)(60,63)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=fh,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=ehi,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, 30404);