#152305

order	512
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
prime radical	C₂⁴	C₂×C₁₆
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

298 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,6)(4,5)(7,8)(9,10,12,18,13,19,26,38,14,20,27,39,28,40,49,60,15,17,25,35,29,36,47,57,30,37,48,58,50,59,65,70)(11,16,22,32,23,33,44,54,24,34,45,55,46,56,64,69,21,31,41,51,42,52,61,66,43,53,62,67,63,68,71,72);(2,5)(6,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,21)(17,31)(18,32)(19,33)(20,34)(25,41)(26,44)(27,45)(28,46)(29,42)(30,43)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(47,61)(48,62)(49,64)(50,63)(57,66)(58,67)(59,68)(60,69)(65,71)(70,72);(1,3,4,7)(2,6,5,8)(10,17)(11,21)(12,25)(19,36)(20,37)(23,42)(24,43)(26,47)(27,48)(32,51)(40,59)(46,63)(49,65)(54,66)(55,67)(69,72)

Polycyclic presentation:

2^9|a^2=f,c^2=d,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=ci,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59501);