#122886

order	512
nilpotency class	4
isoclinism	isoclinic class of #122859
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#331
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₈²	C₂³
prime radical	C₂³	#331
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

248 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=hi,d^2=gi,e^2=fgi,f^2=h,g^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fh,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,g^b=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30082);