#8701666

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3964 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)(3,4)(5,9)(6,7)(8,10)(11,12)(13,14)(15,16)(33,34)(35,42)(36,44)(37,39)(38,41)(40,45)(43,47)(46,48);(2,7)(3,10)(9,16)(11,13)(17,18,23,22)(19,27,29,24)(20,30,28,25)(21,26,31,32)(33,34)(35,42)(36,40)(37,39)(38,41)(43,48)(44,45)(46,47);(17,19,23,29)(18,24,22,27)(20,21,28,31)(25,26,30,32)(33,35,38,37)(34,39,41,42)(36,43,45,46)(40,47,44,48);(1,3)(2,8)(4,7)(5,13)(6,10)(9,12)(11,15)(14,16)(17,20,23,28)(18,25,22,30)(19,21,29,31)(24,26,27,32)(33,34)(35,39)(36,40)(37,42)(38,41)(43,47)(44,45)(46,48);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,13,16,11)(17,21)(18,26)(19,28)(20,29)(22,32)(23,31)(24,30)(25,27)(33,36)(34,40)(35,43)(37,46)(38,45)(39,47)(41,44)(42,48);(1,5,6,15)(2,9,7,16)(3,11,10,13)(4,12,8,14)(17,22,23,18)(19,27,29,24)(20,25,28,30)(21,26,31,32)(33,37,38,35)(34,39,41,42)(36,43,45,46)(40,48,44,47)

Polycyclic presentation:

2^9|b^2=hi,c^2=i,d^2=hi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8608862);