#122912

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	#3169	C₄
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

259 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 78 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)(35,43)(36,47)(37,52)(38,40)(39,45)(41,50)(42,49)(44,59)(46,61)(48,58)(51,62)(53,57)(54,63)(55,60)(56,64);(1,3)(2,7)(4,16)(5,20)(6,13)(8,23)(9,25)(10,12)(11,21)(14,28)(15,17)(18,30)(19,31)(22,26)(24,32)(27,29)(33,35)(34,39)(36,48)(37,53)(38,44)(40,55)(41,57)(42,58)(43,47)(45,50)(46,49)(51,63)(52,54)(56,61)(59,62)(60,64);(3,13)(7,21)(12,23)(14,30)(15,20)(16,31)(22,32)(25,29)(34,40)(35,44)(41,56)(42,52)(45,60)(46,53)(47,62)(48,63)

Polycyclic presentation:

2^9|a^2=e,d^2=gi,e^2=fgh,f^2=h,g^2=i,b^a=bd,c^a=ci,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, 30108);