#10171814

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

3927 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)(3,9,10,4)(5,13,15,8)(11,12,16,14)(17,18)(19,23)(20,24)(21,32)(22,26)(25,29)(27,28)(30,31)(33,34,39,43)(35,44,48,40)(36,37,41,47)(38,42,46,45);(2,7)(4,9)(8,13)(12,14)(17,19,22,28)(18,23,26,27)(20,29,31,21)(24,25,30,32)(33,35)(34,40)(36,45)(37,46)(38,47)(39,48)(41,42)(43,44);(2,7)(3,10)(8,13)(11,16)(17,18,22,26)(19,27,28,23)(20,24,31,30)(21,25,29,32)(33,36,39,41)(34,37,43,47)(35,45,48,42)(38,40,46,44);(1,3,6,10)(2,4,7,9)(5,11,15,16)(8,12,13,14)(17,18)(19,23)(20,30)(21,25)(22,26)(24,31)(27,28)(29,32)(33,34)(35,40)(36,37)(38,42)(39,43)(41,47)(44,48)(45,46);(1,4)(2,3)(5,14)(6,9)(7,10)(8,16)(11,13)(12,15)(17,20,22,31)(18,24,26,30)(19,21,28,29)(23,32,27,25)(33,37)(34,41)(35,46)(36,43)(38,48)(39,47)(40,42)(44,45);(1,5,6,15)(2,8,7,13)(3,11,10,16)(4,12,9,14)(17,21)(18,25)(19,20)(22,29)(23,30)(24,27)(26,32)(28,31)(33,38,39,46)(34,42,43,45)(35,47,48,37)(36,40,41,44)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=hi,e^2=i,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, 10079010);