#232851

order	512
nilpotency class	3
isoclinism	isoclinic class of #232847
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1127 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1127 conjugacy classes of subgroups — too many to draw. Only its 184 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) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|d^2=hi,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gi,d^c=dh,e^c=ei,f^c=fhi,g^c=gi

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, 140047);