#8942627

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	#50059	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=gh,e^2=h,f^2=h,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, 8849823);