#9910899

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3735 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)(3,5)(4,8)(6,9)(7,10)(11,13)(12,14)(15,16)(19,25)(23,29)(26,31)(30,32)(33,34,39,40)(35,36,44,41)(37,42,49,53)(38,51,52,43)(45,47,57,54)(46,55,58,48)(50,62,61,56)(59,63,64,60);(1,3,7,4)(2,8,10,5)(6,11,15,12)(9,14,16,13)(18,22)(23,29)(24,27)(30,32)(34,40)(35,44)(37,49)(43,51)(46,58)(47,54)(50,61)(60,63);(1,3,7,4)(2,5,10,8)(6,11,15,12)(9,13,16,14)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32)(33,35,39,44)(34,41,40,36)(37,45,49,57)(38,46,52,58)(42,54,53,47)(43,55,51,48)(50,59,61,64)(56,63,62,60);(1,4,7,3)(2,8,10,5)(6,12,15,11)(9,14,16,13)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,32)(28,31)(33,36,39,41)(34,35,40,44)(37,47,49,54)(38,48,52,55)(42,45,53,57)(43,46,51,58)(50,60,61,63)(56,59,62,64);(1,5,7,8)(2,3,10,4)(6,14,15,13)(9,12,16,11)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31)(33,37)(34,42)(35,45)(36,47)(38,50)(39,49)(40,53)(41,54)(43,56)(44,57)(46,59)(48,60)(51,62)(52,61)(55,63)(58,64);(1,6,7,15)(2,9,10,16)(3,11,4,12)(5,13,8,14)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,38,39,52)(34,43,40,51)(35,46,44,58)(36,48,41,55)(37,50,49,61)(42,56,53,62)(45,59,57,64)(47,60,54,63)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9818095);