#8595731

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×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3994 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8502927);