#152308

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	#2163
prime radical	C₂⁴	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

296 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 114 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)(5,6,9,17,10,18,29,44,11,14,30,37,31,38,55,61,12,15,28,39,32,40,54,62,8,16,25,41,26,42,52,63)(7,19,22,43,23,45,49,64,24,33,50,56,51,57,66,67,20,13,27,34,46,35,53,59,21,36,47,58,48,60,65,68);(5,7)(6,13)(8,21)(9,27)(10,23)(11,24)(12,20)(14,36)(15,19)(16,33)(17,43)(18,35)(22,28)(25,50)(26,48)(29,53)(30,47)(31,51)(32,46)(34,39)(37,56)(38,60)(40,45)(41,58)(42,57)(44,64)(49,54)(52,66)(55,65)(59,62)(61,67)(63,68);(2,4)(6,14,15,16)(7,20)(9,28)(13,33,19,36)(17,41,39,37)(18,38,40,42)(21,24)(23,46)(25,30)(29,54)(34,58,43,56)(35,57,45,60)(44,63,62,61)(48,51)(52,55)(59,68,64,67)

Polycyclic presentation:

2^9|a^2=f,c^2=d,e^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=ci,f^b=fi,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59504);