#8121440

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

3837 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=gi,e^2=ghi,f^2=g,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, 8028636);