#8924756

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	#48669	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4005 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8831952);