#8750618

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

3815 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)(3,7)(4,12)(5,9)(6,10)(8,13)(11,14)(15,16)(17,18,22,21)(19,24,27,23)(20,28,29,25)(26,30,32,31)(33,34,37,40)(35,38,41,45)(36,43,44,39)(42,48,47,46);(1,3)(2,7)(4,9)(5,12)(6,11)(8,16)(10,14)(13,15)(17,19,22,27)(18,23,21,24)(20,26,29,32)(25,30,28,31)(35,41)(38,45)(42,47)(46,48);(3,11)(4,13)(7,14)(8,12)(17,19,22,27)(18,23,21,24)(20,26,29,32)(25,30,28,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4,6,13)(2,8,10,12)(3,9,11,15)(5,7,16,14)(17,19)(18,24)(20,26)(21,23)(22,27)(25,31)(28,30)(29,32)(33,35,37,41)(34,38,40,45)(36,42,44,47)(39,46,43,48);(1,2)(3,7)(4,8)(5,15)(6,10)(9,16)(11,14)(12,13)(17,20,22,29)(18,25,21,28)(19,26,27,32)(23,30,24,31)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48);(1,5,6,16)(2,9,10,15)(3,12,11,8)(4,14,13,7)(17,21,22,18)(19,24,27,23)(20,25,29,28)(26,30,32,31)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8657814);