#214049

order	512
nilpotency class	3
isoclinism	isoclinic class of #214040
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

753 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)(3,10,5,8)(4,15,18,28)(6,12)(7,11)(9,17,25,20)(13,29)(14,26,19,23)(16,32)(21,27)(22,30)(24,31)(33,34,37,42,38,41,50,55)(35,39,45,53,44,52,60,63)(36,46,47,56,48,40,49,54)(43,57,59,64,58,51,61,62);(1,3,7,5)(2,8,11,10)(4,13,18,16)(6,19,21,14)(9,24,25,22)(12,26,27,23)(15,32,28,29)(17,31,20,30)(33,35)(34,39)(36,43)(37,45)(38,44)(40,51)(41,52)(42,53)(46,57)(47,59)(48,58)(49,61)(50,60)(54,62)(55,63)(56,64);(1,4)(2,9)(3,13)(5,16)(6,20)(7,18)(8,22)(10,24)(11,25)(12,28)(14,31)(15,27)(17,21)(19,30)(23,29)(26,32)(33,36,38,48)(34,40,41,46)(35,43,44,58)(37,49,50,47)(39,51,52,57)(42,56,55,54)(45,61,60,59)(53,64,63,62);(1,5,7,3)(2,10,11,8)(4,16,18,13)(6,19,21,14)(9,24,25,22)(12,26,27,23)(15,29,28,32)(17,31,20,30)(35,44)(39,52)(43,58)(45,60)(51,57)(53,63)(59,61)(62,64)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,f^b=fh,g^b=gh,f^c=fh,g^c=ghi,g^d=gh

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, 121245);