#10168352

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 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)(3,8)(4,5)(6,14)(7,9)(10,16)(11,13)(12,15)(17,18)(19,27)(20,24)(21,30)(22,26)(23,29)(25,32)(28,31)(34,40)(36,46)(41,43)(44,45);(1,3,7,4)(2,8,9,5)(6,11,16,12)(10,15,14,13)(17,18,22,26)(19,23,29,27)(20,28,31,24)(21,25,32,30)(33,34)(35,41)(36,42)(37,43)(38,44)(39,40)(45,48)(46,47);(2,9)(3,4)(10,14)(11,12)(17,19,22,29)(18,23,26,27)(20,25,31,30)(21,28,32,24)(33,35,39,37)(34,41,40,43)(36,45,46,44)(38,42,48,47);(1,4,7,3)(2,5,9,8)(6,12,16,11)(10,13,14,15)(17,19)(18,23)(20,30)(21,28)(22,29)(24,32)(25,31)(26,27)(33,36)(34,42)(35,45)(37,44)(38,43)(39,46)(40,47)(41,48);(1,5)(2,4)(3,9)(6,15)(7,8)(10,11)(12,14)(13,16)(17,20)(18,24)(19,25)(21,23)(22,31)(26,28)(27,32)(29,30)(33,37,39,35)(34,43,40,41)(36,45,46,44)(38,42,48,47);(1,6,7,16)(2,10,9,14)(3,11,4,12)(5,13,8,15)(17,21)(18,25)(19,28)(20,27)(22,32)(23,31)(24,29)(26,30)(33,38,39,48)(34,44,40,45)(35,42,37,47)(36,43,46,41)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=i,e^2=ghi,f^2=gh,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, 10075548);