#114238

order	512
nilpotency class	3
isoclinism	isoclinic class of #114229
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

742 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)(3,13,17,8)(4,18)(5,12)(6,11)(7,10)(9,20)(14,26,31,32)(15,24,16,25)(19,23,30,29)(21,27)(22,28)(33,34,38,42,39,44,37,43)(35,45,47,40,48,55,53,56)(36,41,51,58,52,59,50,60)(46,57,61,54,62,64,49,63);(1,3)(2,8)(4,19)(5,16)(6,15)(7,17)(9,26)(10,13)(11,25)(12,24)(14,21)(18,29)(20,31)(22,30)(23,28)(27,32)(33,35,39,48)(34,40,44,56)(36,49,52,61)(37,53,38,47)(41,57,59,64)(42,55,43,45)(46,50,62,51)(54,60,63,58);(1,4,7,22)(2,9,10,27)(3,14,17,31)(5,21,6,20)(8,23,13,29)(11,28,12,18)(15,19,16,30)(24,26,25,32)(33,36,39,52)(34,41,44,59)(35,46,48,62)(37,50,38,51)(40,54,56,63)(42,58,43,60)(45,57,55,64)(47,61,53,49)

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,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,e^c=ehi,f^c=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21434);