#117248

order	512
nilpotency class	3
isoclinism	isoclinic class of #117239
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

507 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)(3,13,17,23,16,24,15,8)(4,9,19,27,20,28,21,26)(14,29,18,22,30,25,31,32)(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);(1,3,5,15,7,16,6,17)(2,8,10,13,12,23,11,24)(4,18,19,30,20,31,21,14)(9,25,26,22,28,29,27,32)(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)(2,9)(3,14)(5,21)(6,19)(7,20)(8,22)(10,27)(11,26)(12,28)(13,25)(15,31)(16,30)(17,18)(23,32)(24,29)(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=i,d^2=h,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,f^c=fhi,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24444);