#152827

order	512
nilpotency class	5
isoclinism	isoclinic class of #151115
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

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

normal subgroups only

This group has 210 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60023);