#8946238

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8853434);