#10247199

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,d^2=hi,e^2=gh,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, 10154395);