#500304

order	512
nilpotency class	4
isoclinism	isoclinic class of #499949
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

752 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 152 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 — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=eh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^c=ei,f^c=fhi,g^c=gi,h^c=hi,f^d=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, 407500);