#194624

order	512
nilpotency class	3
isoclinism	isoclinic class of #19050
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

963 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)(3,12,15,23)(4,16)(5,10)(6,20)(7,11)(8,14,24,19)(9,18)(13,28,29,22)(17,26)(21,25)(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);(1,3,5,19,7,15,6,14)(2,8,10,23,11,24,20,12)(4,13,18,31,17,29,21,30)(9,22,26,32,25,28,16,27)(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)(2,9)(3,13)(5,18)(6,21)(7,17)(8,22)(10,26)(11,25)(12,27)(14,30)(15,29)(16,20)(19,31)(23,32)(24,28)(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);(4,17)(9,25)(13,29)(16,26)(18,21)(22,28)(27,32)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101820);