#194612

order	512
nilpotency class	3
isoclinism	isoclinic class of #18992
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

954 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)(3,12,15,24)(4,16)(5,11)(6,20)(7,10)(8,14,23,19)(9,17)(13,28,31,22)(18,26)(21,25)(27,29,32,30)(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)(2,8,11,24,10,23,20,12)(4,13,17,30,18,31,21,29)(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);(1,4,7,18)(2,9,10,25)(3,13,15,31)(5,17,6,21)(8,22,23,28)(11,26,20,16)(12,27,24,32)(14,29,19,30)(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);(2,10)(8,23)(9,25)(11,20)(12,24)(16,26)(22,28)(27,32)

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,d^a=dh,f^a=fh,e^b=eh,f^b=fi,g^b=gh,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, 101808);