#10247186

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3741 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)(3,11)(4,9)(5,15)(6,7)(8,10)(12,16)(13,14)(33,34,38,39)(35,43,46,40)(36,41,47,48)(37,50,52,42)(44,57,59,53)(45,54,62,58)(49,56,51,55)(60,63,61,64);(1,3,8,6)(2,7,10,11)(4,12,5,13)(9,14,15,16)(17,18,22,25)(19,20,27,23)(21,24,30,32)(26,28,29,31)(34,39)(36,47)(40,43)(42,50)(44,59)(49,51)(54,58)(60,61);(1,4,8,5)(2,9,10,15)(3,12,6,13)(7,14,11,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,35,38,46)(34,40,39,43)(36,44,47,59)(37,45,52,62)(41,53,48,57)(42,54,50,58)(49,60,51,61)(55,63,56,64);(1,5,8,4)(2,9,10,15)(3,12,6,13)(7,16,11,14)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,32,29)(33,34)(35,43)(36,48)(37,42)(38,39)(40,46)(41,47)(44,53)(45,58)(49,56)(50,52)(51,55)(54,62)(57,59)(60,63)(61,64);(1,6,8,3)(2,7,10,11)(4,13,5,12)(9,14,15,16)(17,19)(18,20)(21,29)(22,27)(23,25)(24,31)(26,30)(28,32)(33,36)(34,41)(35,44)(37,51)(38,47)(39,48)(40,53)(42,56)(43,57)(45,61)(46,59)(49,52)(50,55)(54,64)(58,63)(60,62);(1,7)(2,6)(3,10)(4,14)(5,16)(8,11)(9,13)(12,15)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,37)(34,42)(35,45)(36,49)(38,52)(39,50)(40,54)(41,55)(43,58)(44,60)(46,62)(47,51)(48,56)(53,63)(57,64)(59,61)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=ghi,e^2=gh,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, 10154382);