#6343690

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343678
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2625 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=h,f^2=i,b^a=bf,e^a=ei,f^a=fh,f^b=fi,g^b=ghi,d^c=di,e^c=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250886);