#118520

order	512
nilpotency class	3
isoclinism	isoclinic class of #118515
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
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

530 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,e^b=eh,f^b=fhi,d^c=di,e^c=ei,f^c=fh,g^c=gi

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, 25716);