#8119361

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

4054 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)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(21,29)(24,31)(27,28)(30,32)(33,34)(35,38)(36,41)(37,47)(39,42)(40,45)(43,48)(44,46);(2,7)(3,10)(9,12)(11,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,35)(34,40)(36,43)(37,44)(38,42)(39,45)(41,46)(47,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,31,32)(33,36,39,37)(34,41,42,47)(35,43,45,44)(38,48,40,46);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,20,22,19)(18,23,25,26)(21,28,29,27)(24,30,31,32)(33,37,39,36)(34,41,42,47)(35,44,45,43)(38,48,40,46);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,38,39,40)(34,35,42,45)(36,46,37,48)(41,44,47,43);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,35)(34,40)(36,43)(37,44)(38,42)(39,45)(41,46)(47,48)

Polycyclic presentation:

2^9|c^2=gi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8026557);