#9054436

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3645 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:

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

Polycyclic presentation:

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

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, 8961632);