#114339

order	512
nilpotency class	3
isoclinism	isoclinic class of #114335
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

726 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 130 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=hi,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=ehi,f^c=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, 21535);