#10142138

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

4080 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10049334);