#10195494

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

4043 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,d^2=h,e^2=g,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, 10102690);