#339894

order	512
nilpotency class	3
isoclinism	isoclinic class of #339847
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#560
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

497 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 247090);