#152293

order	512
nilpotency class	5
isoclinism	isoclinic class of #13697
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₁₆
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×#366
prime radical	C₂⁴	Q₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

238 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,d^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ci

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59489);