#10247191

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

3715 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10154387);