#114343

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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

732 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=hi,f^2=h,g^2=i,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, 21539);