#122904

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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=i,d^2=gi,e^2=fgh,f^2=h,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gh,c^b=ch,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, 30100);