#194616

order	512
nilpotency class	3
isoclinism	isoclinic class of #19014
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

961 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fhi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101812);