#10246838

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	#52634	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3994 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,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, 10154034);