#347433

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	#62978	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1922 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)(3,13,4,9)(5,10,19,27)(6,20,22,11)(8,17,24,15)(14,25,16,29)(18,28,21,26)(23,31,32,30)(35,43)(37,48)(39,51)(40,54)(41,49)(44,57)(45,50)(47,59)(52,63)(53,56)(58,61)(62,64);(1,3,7,4)(2,8,12,24)(5,14,19,16)(6,17,22,15)(9,20,13,11)(10,23,27,32)(18,31,21,30)(25,26,29,28)(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);(1,4,7,3)(2,9,12,13)(5,16,19,14)(6,17,22,15)(8,20,24,11)(10,25,27,29)(18,31,21,30)(23,26,32,28)(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);(1,5,7,19)(2,10,12,27)(3,14,4,16)(6,21,22,18)(8,23,24,32)(9,25,13,29)(11,28,20,26)(15,31,17,30)(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,b^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, 254629);