#128481

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	#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

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

normal subgroups only

This group has 318 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:

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

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,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

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343150, #72343182, #72343187
maximal normal subgroup of	#72343150, #72343182, #72343187

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35677);