#10195951

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

4155 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=gh,e^2=gh,f^2=hi,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, 10103147);