#8709119

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	#40818	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4149 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)(3,10)(4,14)(5,8)(6,11)(7,9)(12,16)(13,15)(33,34)(35,43)(36,40)(37,39)(38,42)(41,46)(44,48)(45,47);(1,3,7,5)(2,8,9,10)(4,15,11,12)(6,13,14,16)(17,18,22,23)(19,21,25,30)(20,24,28,31)(26,29,32,27)(33,35,38,37)(34,39,42,43)(36,44,46,45)(40,47,41,48);(2,9)(4,11)(8,10)(12,15)(18,23)(19,25)(24,31)(26,32)(33,36,38,46)(34,40,42,41)(35,44,37,45)(39,47,43,48);(1,4)(2,6)(3,12)(5,15)(7,11)(8,16)(9,14)(10,13)(17,19)(18,21)(20,26)(22,25)(23,30)(24,29)(27,31)(28,32)(33,36)(34,41)(35,44)(37,45)(38,46)(39,48)(40,42)(43,47);(1,5,7,3)(2,10,9,8)(4,15,11,12)(6,13,14,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)(33,37,38,35)(34,39,42,43)(36,45,46,44)(40,47,41,48);(1,6)(2,11)(3,13)(4,9)(5,16)(7,14)(8,12)(10,15)(17,21)(18,25)(19,23)(20,27)(22,30)(24,26)(28,29)(31,32)(33,34)(35,39)(36,40)(37,43)(38,42)(41,46)(44,47)(45,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8616315);