#8905193

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	#47991	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4211 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^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, 8812389);