#128482

order	512
nilpotency class	4
isoclinism	isoclinic class of #11522
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	#525	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

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

normal subgroups only

This group has 311 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319273, #72343187
maximal normal subgroup of	#72319273, #72343187

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35678);