#152979

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×D₃₂	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 284 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)(5,6,11,18,14,20,13,42)(7,21,25,22,27,38,17,28)(8,23,31,54,34,56,33,47)(9,35,10,26,40,57,39,44)(12,24,37,36,15,43,41,19)(16,46,48,63,50,29,59,30)(32,65,61,68,45,66,64,55)(49,58,53,51,60,52,67,62);(1,2,3,4)(5,7,22,20,12,26,9,36,11,17,38,18,37,35,10,24,14,27,28,6,15,44,40,19,13,25,21,42,41,57,39,43)(8,23,52,50,32,55,29,60,31,47,62,48,61,66,30,53,34,56,58,16,45,68,46,49,33,54,51,59,64,65,63,67);(5,8,28,58,12,32,40,46,11,31,21,51,37,61,39,63,14,34,22,52,15,45,9,29,13,33,38,62,41,64,10,30)(6,16,26,55,19,49,17,47,42,59,35,66,43,67,27,56,20,50,44,68,36,60,25,54,18,48,57,65,24,53,7,23)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=eg,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, 60175);