#8119581

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3993 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)(3,10)(4,12)(5,7)(6,8)(9,14)(11,15)(13,16)(17,18)(19,20)(21,24)(22,29)(23,25)(26,31)(27,32)(28,30)(34,37)(39,42)(40,45)(47,48);(1,3,7,6)(2,8,5,10)(4,13,14,11)(9,16,12,15)(17,19,24,23)(18,25,21,20)(22,27,31,30)(26,32,29,28)(33,34,38,37)(35,39,43,42)(36,40,46,45)(41,47,44,48);(2,5)(8,10)(9,12)(15,16)(17,20)(18,23)(19,21)(22,28)(24,25)(26,30)(27,29)(31,32)(33,35,38,43)(34,39,37,42)(36,44,46,41)(40,48,45,47);(1,4,7,14)(2,9,5,12)(3,11,6,13)(8,15,10,16)(17,21,24,18)(19,25,23,20)(22,29,31,26)(27,32,30,28)(33,34,38,37)(35,39,43,42)(36,40,46,45)(41,47,44,48);(1,5,7,2)(3,10,6,8)(4,12,14,9)(11,16,13,15)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,36,38,46)(34,40,37,45)(35,41,43,44)(39,47,42,48);(1,6,7,3)(2,10,5,8)(4,13,14,11)(9,16,12,15)(17,23,24,19)(18,25,21,20)(22,30,31,27)(26,32,29,28)(33,37,38,34)(35,42,43,39)(36,45,46,40)(41,48,44,47)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=g,e^2=gi,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, 8026777);