#152843

order	512
nilpotency class	5
isoclinism	isoclinic class of #151127
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2177
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#509
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

208 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 74 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)(5,6,11,18,13,21,43,20)(7,22,25,15,28,49,27,51)(8,29,31,57,34,14,33,47)(9,35,37,17,39,55,67,56)(10,40,38,16,23,53,61,54)(12,44,42,19,45,52,36,58)(24,48,62,68,41,65,32,59)(26,50,30,66,64,46,63,60);(5,7,23,24,12,26,9,34,43,25,38,62,36,30,67,33,13,28,10,41,45,64,39,8,11,27,61,32,42,63,37,31)(6,14,17,50,19,48,54,22,18,57,55,60,52,59,40,51,21,29,56,46,58,65,16,49,20,47,35,66,44,68,53,15);(2,4)(5,8,9,30,36,24,10,27,11,33,37,26,12,32,38,28,13,34,39,63,42,41,23,25,43,31,67,64,45,62,61,7)(6,15,16,48,52,60,17,29,20,49,40,68,19,50,35,57,21,51,54,65,44,66,56,14,18,22,53,59,58,46,55,47)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60039);