#332023

order	512
nilpotency class	3
isoclinism	isoclinic class of #331999
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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

489 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=i,e^2=h,f^2=i,g^2=hi,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=ghi,f^c=fi,g^c=ghi,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, 239219);