#10408139

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

4336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,9)(4,6)(5,12)(7,11)(8,14)(10,15)(13,16)(18,23)(19,25)(21,29)(28,31)(33,34)(35,37)(36,40)(38,41)(39,42)(43,45)(44,46)(47,48);(2,4)(7,9)(8,12)(15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,48)(46,47);(1,3)(2,7)(4,9)(5,10)(6,11)(8,15)(12,16)(13,14)(17,19)(18,20)(21,27)(22,25)(23,24)(26,28)(29,32)(30,31)(33,36,39,37)(34,40,42,35)(38,44,47,45)(41,46,48,43);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(33,35)(34,36)(37,42)(38,46)(39,40)(41,45)(43,47)(44,48);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27)(33,37,39,36)(34,40,42,35)(38,45,47,44)(41,46,48,43);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,46)(42,48)

Polycyclic presentation:

2^9|c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10315335);