#152872

order	512
nilpotency class	5
isoclinism	isoclinic class of #13709
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2226
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×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

190 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 94 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)(7,19,23,52)(8,17)(9,28)(10,33)(11,35)(12,18)(13,21,39,58)(14,26)(15,31)(16,34)(20,54,56,68)(22,53,24,51)(25,60)(27,47)(29,50,65,46)(30,42)(32,63)(36,43,55,49)(37,64,40,66)(38,59,41,57)(44,62)(45,61)(48,67);(5,7,10,22,12,23,34,24)(6,13,16,38,18,39,33,41)(8,20,26,55,27,56,11,36)(9,29,30,64,32,65,67,66)(14,37,17,46,35,40,47,50)(15,43,44,68,45,49,60,54)(19,48,51,63,52,42,53,28)(21,25,57,61,58,62,59,31);(1,2,3,4)(5,8,10,26,12,27,34,11)(6,14,16,17,18,35,33,47)(7,20,22,55,23,56,24,36)(9,25,30,61,32,62,67,31)(13,37,38,46,39,40,41,50)(15,42,44,28,45,48,60,63)(19,49,51,54,52,43,53,68)(21,29,57,64,58,65,59,66)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60068);