#347425

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	#62977	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1921 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)(3,13,17,21)(4,10,19,26)(5,8,6,11)(9,15,24,16)(14,28,30,27)(18,22,20,25)(23,31,32,29)(35,43)(37,48)(39,51)(40,54)(41,49)(44,57)(45,50)(47,59)(52,63)(53,56)(58,61)(62,64);(2,8)(5,6)(9,21)(10,22)(11,12)(13,24)(15,16)(18,20)(23,28)(25,26)(27,32)(29,31)(33,34)(35,39)(36,40)(37,49)(38,42)(41,48)(43,56)(44,52)(45,53)(46,54)(47,60)(50,51)(55,59)(57,63)(58,64)(61,62);(1,3,7,17)(2,9,12,24)(4,14,19,30)(5,16,6,15)(8,21,11,13)(10,23,26,32)(18,31,20,29)(22,28,25,27)(33,35)(34,39)(36,44)(37,50)(38,45)(40,52)(41,56)(42,53)(43,48)(46,58)(47,61)(49,51)(54,64)(55,63)(57,59)(60,62);(1,4)(2,10)(3,14)(5,18)(6,20)(7,19)(8,22)(9,23)(11,25)(12,26)(13,27)(15,29)(16,31)(17,30)(21,28)(24,32)(33,36,38,46)(34,40,42,54)(35,44,45,58)(37,47,48,59)(39,52,53,64)(41,55,49,60)(43,57,50,61)(51,62,56,63)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=hi,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, 254621);