#128484

order	512
nilpotency class	4
isoclinism	isoclinic class of #11521
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#410
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	#8160
prime radical	#532	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

313 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,e^2=h,h^2=i,b^a=bd,c^a=ci,d^a=di,f^a=fg,c^b=ch,d^b=dg,e^b=ei,g^b=gi,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319273, #72343182, #72343184
maximal normal subgroup of	#72319273, #72343182, #72343184

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35680);