#152980

order	512
nilpotency class	5
isoclinism	isoclinic class of #13700
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2215
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

274 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=egi,d^2=ghi,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,h^a=hi,d^b=di,f^b=fi

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, 60176);