#8678990

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

3960 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8586186);