#118250

order	512
nilpotency class	3
isoclinism	isoclinic class of #118246
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

515 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)(3,11)(4,15,17,8)(5,18)(6,10)(7,13)(9,19)(12,24,27,32)(14,25)(16,23,30,29)(20,31,22,26)(21,28)(33,34,37,42,38,41,49,55)(35,39,45,51,44,52,57,62)(36,40,48,54,47,53,60,63)(43,50,56,61,46,58,59,64);(1,3,5,13,6,14,19,28)(2,7,9,11,10,21,18,25)(4,12,16,26,17,27,30,31)(8,22,23,32,15,20,29,24)(33,35)(34,39)(36,46,47,43)(37,45)(38,44)(40,50,53,58)(41,52)(42,51)(48,59,60,56)(49,57)(54,61,63,64)(55,62);(1,4,6,17)(2,8,10,15)(3,12,14,27)(5,16,19,30)(7,20,21,22)(9,23,18,29)(11,24,25,32)(13,26,28,31)(33,36)(34,40)(35,43)(37,48)(38,47)(39,50)(41,53)(42,54)(44,46)(45,56)(49,60)(51,61)(52,58)(55,63)(57,59)(62,64)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,e^b=eh,f^b=fi,d^c=di,g^c=gh

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, 25446);