#10216931

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4014 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,d^2=hi,e^2=gi,f^2=gi,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, 10124127);