#10143686

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 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)(3,8)(4,10)(5,11)(6,14)(7,9)(12,15)(13,16)(17,18,21,23)(19,25,27,22)(20,24,30,29)(26,32,28,31)(33,34)(35,45)(36,42)(37,38)(39,43)(40,44)(41,47)(46,48);(1,3,7,12)(2,8,9,15)(4,13,14,5)(6,11,10,16)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,30)(29,31)(33,35)(34,41)(36,39)(37,46)(38,43)(40,47)(42,48)(44,45);(2,9)(3,12)(4,14)(11,16)(18,23)(22,25)(24,29)(31,32)(33,36,40,37)(34,42,44,38)(35,39,47,46)(41,48,45,43);(1,4)(2,10)(3,13)(5,12)(6,9)(7,14)(8,16)(11,15)(17,18)(19,22)(20,29)(21,23)(24,30)(25,27)(26,32)(28,31)(33,37,40,36)(34,42,44,38)(35,46,47,39)(41,48,45,43);(1,5,7,13)(2,11,9,16)(3,14,12,4)(6,15,10,8)(17,20)(18,24)(19,26)(21,30)(22,31)(23,29)(25,32)(27,28)(33,38,40,42)(34,37,44,36)(35,43,47,48)(39,41,46,45);(1,6,7,10)(2,4,9,14)(3,11,12,16)(5,8,13,15)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,30)(29,31)(33,39,40,46)(34,43,44,48)(35,36,47,37)(38,45,42,41)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=ghi,d^2=ghi,e^2=gh,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, 10050882);