#249461

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1054 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1054 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=hi,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, 156657);