#251778

order	512
nilpotency class	3
isoclinism	isoclinic class of #251761
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

689 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 689 conjugacy classes of subgroups — too many to draw. Only its 134 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=i,d^2=i,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,d^b=di,d^c=di,e^c=eh,f^c=fi,g^c=gh,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 158974);