#118952

order	512
nilpotency class	3
isoclinism	isoclinic class of #118950
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#117
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

490 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,e^c=eh,f^c=fi

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, 26148);