#8757322

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

3707 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)(3,9)(4,11)(5,13)(6,8)(7,14)(10,12)(15,16)(18,23)(21,29)(24,30)(28,32)(33,34,39,36)(35,43,44,40)(37,47,50,41)(38,48,51,42)(45,52,57,55)(46,53,58,56)(49,54,61,60)(59,63,64,62);(1,2,6,8)(3,9,11,4)(5,7,14,13)(10,15,16,12)(20,27)(24,30)(26,31)(28,32)(35,44)(38,51)(40,43)(42,48)(45,57)(49,61)(52,55)(54,60);(1,3)(2,4)(5,10)(6,11)(7,12)(8,9)(13,15)(14,16)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,35)(34,40)(36,43)(37,45)(38,46)(39,44)(41,52)(42,53)(47,55)(48,56)(49,59)(50,57)(51,58)(54,62)(60,63)(61,64);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,19,22,25)(18,21,23,29)(20,26,27,31)(24,28,30,32)(33,36,39,34)(35,40,44,43)(37,47,50,41)(38,48,51,42)(45,52,57,55)(46,53,58,56)(49,60,61,54)(59,62,64,63);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,14)(15,16)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,37,39,50)(34,41,36,47)(35,45,44,57)(38,49,51,61)(40,52,43,55)(42,54,48,60)(46,59,58,64)(53,62,56,63);(1,5,6,14)(2,7,8,13)(3,10,11,16)(4,12,9,15)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,38)(34,42)(35,46)(36,48)(37,49)(39,51)(40,53)(41,54)(43,56)(44,58)(45,59)(47,60)(50,61)(52,62)(55,63)(57,64)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=gh,e^2=gi,f^2=g,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, 8664518);