#122864

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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=hi,d^2=gi,e^2=fg,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, 30060);