#347419

order	512
nilpotency class	3
isoclinism	isoclinic class of #347034
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62976	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2033 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)(3,8,14,24)(4,9,16,26)(5,17,20,10)(7,18,22,13)(12,28,30,23)(15,27,19,25)(21,31,32,29)(33,34)(35,39)(36,41)(37,49)(38,43)(40,48)(42,52)(44,60)(45,59)(46,56)(47,58)(50,55)(51,57)(53,62)(54,61)(63,64);(1,3)(2,7)(4,12)(5,18)(6,14)(8,17)(9,21)(10,24)(11,22)(13,20)(15,31)(16,30)(19,29)(23,25)(26,32)(27,28)(33,35)(34,39)(36,44)(37,50)(38,46)(40,53)(41,54)(42,59)(43,56)(45,52)(47,61)(48,63)(49,55)(51,62)(57,64)(58,60);(1,3)(2,8)(4,12)(5,13)(6,14)(7,17)(9,23)(10,22)(11,24)(15,29)(16,30)(18,20)(19,31)(21,25)(26,28)(27,32)(33,36)(34,40)(35,44)(37,51)(38,47)(39,53)(41,49)(42,58)(43,57)(45,63)(46,61)(48,52)(50,62)(54,55)(56,64)(59,60);(1,4,6,16)(2,9,11,26)(3,12,14,30)(5,19,20,15)(7,21,22,32)(8,23,24,28)(10,27,17,25)(13,31,18,29)(33,36)(34,41)(35,44)(37,48)(38,47)(39,54)(40,49)(42,57)(43,58)(45,62)(46,61)(50,63)(51,52)(53,55)(56,60)(59,64)

Polycyclic presentation:

2^9|a^2=h,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,e^b=eh,f^b=fi,g^b=ghi,e^c=ei,f^c=fi,g^c=gh,e^d=ehi,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254615);