#8595959

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3966 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=g,e^2=gh,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, 8503155);