#195397

order	512
nilpotency class	3
isoclinism	isoclinic class of #18414
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

967 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)(33,34,37,42,38,41,50,55)(35,39,45,53,44,52,59,63)(36,46,48,56,47,40,49,54)(43,51,58,62,57,61,60,64);(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)(33,35,37,45,38,44,50,59)(34,39,42,53,41,52,55,63)(36,43,49,60,47,57,48,58)(40,51,56,64,46,61,54,62);(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)(34,40)(35,43)(37,49)(38,47)(39,51)(41,46)(42,56)(44,57)(45,60)(48,50)(52,61)(53,64)(54,55)(58,59)(62,63);(3,14)(8,23)(12,24)(13,29)(15,19)(22,28)(27,32)(30,31)(34,41)(39,52)(40,46)(42,55)(51,61)(53,63)(54,56)(62,64)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fh,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, 102593);