#10316824

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3728 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,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, 10224020);