#113610

order	512
nilpotency class	3
isoclinism	isoclinic class of #113605
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

755 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)(3,13,16,8)(4,17)(5,12)(6,11)(7,10)(9,20)(14,30,31,25)(15,24,22,28)(18,29,32,23)(19,26)(21,27)(33,34,38,42,39,43,37,53)(35,44,46,40,47,57,54,56)(36,48,51,60,52,41,50,59)(45,58,49,55,62,61,63,64);(1,3)(2,8)(4,18)(5,15)(6,22)(7,16)(9,25)(10,13)(11,24)(12,28)(14,21)(17,29)(19,32)(20,31)(23,26)(27,30)(33,35)(34,40)(36,49)(37,54)(38,46)(39,47)(41,58)(42,57)(43,56)(44,53)(45,50)(48,61)(51,62)(52,63)(55,59)(60,64);(1,4,7,19)(2,9,10,27)(3,14,16,31)(5,21,6,20)(8,23,13,29)(11,26,12,17)(15,32,22,18)(24,30,28,25)(33,36,39,52)(34,41,43,48)(35,45,47,62)(37,51,38,50)(40,55,56,64)(42,60,53,59)(44,61,57,58)(46,63,54,49)

Polycyclic presentation:

2^9|a^2=g,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=eh,f^a=fh,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=dh,e^c=ehi,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20806);