#117452

order	512
nilpotency class	3
isoclinism	isoclinic class of #117443
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	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=hi,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,e^b=eh,f^b=fi,d^c=dh,e^c=ehi,f^c=fhi,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, 24648);