#8687323

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3816 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8594519);