#249459

order	512
nilpotency class	3
isoclinism	isoclinic class of #249427
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1015 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1015 conjugacy classes of subgroups — too many to draw. Only its 148 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156655);