#8429497

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

3943 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=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, 8336693);