#195787

order	512
nilpotency class	3
isoclinism	isoclinic class of #18810
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

968 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)(3,13,15,23)(4,18,21,26)(5,11)(6,12)(7,10)(8,16,24,17)(9,20,25,19)(14,28)(22,29)(27,30)(31,32)(33,34,38,43,39,42,37,51)(35,40,46,55,47,54,45,61)(36,48,49,41,50,57,52,56)(44,58,59,53,60,64,62,63);(1,3,5,17,7,15,6,16)(2,8,11,23,10,24,12,13)(4,14,20,30,21,29,19,31)(9,22,18,32,25,28,26,27)(33,35,38,46,39,47,37,45)(34,40,43,55,42,54,51,61)(36,44,49,59,50,60,52,62)(41,53,57,64,56,63,48,58);(1,4,7,21)(2,9,10,25)(3,14,15,29)(5,20,6,19)(8,22,24,28)(11,18,12,26)(13,27,23,32)(16,31,17,30)(33,36)(34,41)(35,44)(37,52)(38,49)(39,50)(40,53)(42,56)(43,57)(45,62)(46,59)(47,60)(48,51)(54,63)(55,64)(58,61);(2,10)(3,15)(9,25)(11,12)(14,29)(16,17)(18,26)(30,31)(34,42)(40,54)(41,56)(43,51)(48,57)(53,63)(55,61)(58,64)

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=dhi,f^a=fi,d^b=dh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fhi,g^c=gh

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, 102983);