#347420

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	#62974	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1977 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)(33,34,39,44)(35,45,49,53)(36,42,51,58)(37,40,38,43)(41,47,56,48)(46,60,62,59)(50,54,52,57)(55,63,64,61);(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)(34,40)(37,38)(41,53)(42,54)(43,44)(45,56)(47,48)(50,52)(55,60)(57,58)(59,64)(61,63);(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)(33,35)(34,41)(36,46)(37,48)(38,47)(39,49)(40,53)(42,55)(43,45)(44,56)(50,63)(51,62)(52,61)(54,60)(57,59)(58,64);(1,4)(2,9)(3,12)(5,19)(6,16)(7,21)(8,23)(10,27)(11,26)(13,31)(14,30)(15,20)(17,25)(18,29)(22,32)(24,28)(33,36,39,51)(34,42,44,58)(35,46,49,62)(37,50,38,52)(40,54,43,57)(41,55,56,64)(45,59,53,60)(47,61,48,63)

Polycyclic presentation:

2^9|a^2=h,d^2=i,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, 254616);