#8618301

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

4078 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,d^2=gh,e^2=gi,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, 8525497);