#8119216

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

3848 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)(8,15)(11,12)(14,16)(17,18,24,23)(19,28,22,25)(20,27,21,26)(29,31,30,32)(33,34,39,43)(35,42,38,40)(36,37,41,48)(44,47,46,45);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,19,24,22)(18,25,23,28)(20,30,21,29)(26,32,27,31)(33,35,39,38)(34,40,43,42)(36,44,41,46)(37,45,48,47);(1,3,6,4)(2,7,9,10)(5,11,13,12)(8,14,15,16)(17,20,24,21)(18,26,23,27)(19,29,22,30)(25,31,28,32)(33,36)(34,37)(35,44)(38,46)(39,41)(40,45)(42,47)(43,48);(1,4,6,3)(2,7,9,10)(5,12,13,11)(8,14,15,16)(17,21,24,20)(18,27,23,26)(19,30,22,29)(25,32,28,31)(33,35,39,38)(34,40,43,42)(36,44,41,46)(37,45,48,47);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,22,24,19)(18,28,23,25)(20,30,21,29)(26,32,27,31)(33,37)(34,41)(35,45)(36,43)(38,47)(39,48)(40,46)(42,44);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,15)(10,16)(17,23,24,18)(19,28,22,25)(20,27,21,26)(29,32,30,31)(33,38,39,35)(34,42,43,40)(36,46,41,44)(37,47,48,45)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=ghi,e^2=h,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, 8026412);