#331952

order	512
nilpotency class	3
isoclinism	isoclinic class of #331927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3207	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

507 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239148);