#8739387

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

3739 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=gh,e^2=h,f^2=gh,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, 8646583);