#8284452

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4083 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8191648);