#8757343

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3709 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=g,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664539);