#10159208

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

3980 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)(3,12)(4,5)(6,16)(7,9)(8,13)(10,15)(11,14)(17,18)(19,21)(20,24)(22,26)(23,30)(25,31)(27,32)(28,29)(33,34,39,44)(35,43,38,40)(36,46,47,41)(37,42,48,45);(1,3,7,13)(2,8,9,12)(4,14,15,6)(5,16,10,11)(17,19,22,25)(18,21,26,31)(20,28,30,27)(23,32,24,29)(33,35,39,38)(34,40,44,43)(36,45,47,42)(37,46,48,41);(2,9)(3,13)(4,15)(11,16)(17,20,22,30)(18,23,26,24)(19,27,25,28)(21,29,31,32)(33,36,39,47)(34,41,44,46)(35,45,38,42)(37,43,48,40);(1,4)(2,10)(3,14)(5,9)(6,13)(7,15)(8,11)(12,16)(17,20)(18,24)(19,28)(21,29)(22,30)(23,26)(25,27)(31,32)(33,36)(34,41)(35,45)(37,40)(38,42)(39,47)(43,48)(44,46);(1,5,7,10)(2,4,9,15)(3,11,13,16)(6,12,14,8)(17,19)(18,21)(20,27)(22,25)(23,29)(24,32)(26,31)(28,30)(33,37)(34,42)(35,41)(36,43)(38,46)(39,48)(40,47)(44,45);(1,6,7,14)(2,11,9,16)(3,4,13,15)(5,12,10,8)(17,21)(18,25)(19,26)(20,29)(22,31)(23,28)(24,27)(30,32)(33,38,39,35)(34,43,44,40)(36,42,47,45)(37,41,48,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,e^2=ghi,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, 10066404);