#8944476

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3930 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)(3,9)(4,5)(6,14)(7,8)(10,15)(11,13)(12,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,32)(29,30)(33,34,38,41)(35,39,45,48)(36,40,46,37)(42,43,44,47);(1,3,7,4)(2,5,8,9)(6,11,15,12)(10,13,14,16)(17,19,22,25)(18,23,21,27)(20,26,30,28)(24,31,29,32)(33,35)(34,39)(36,44)(37,43)(38,45)(40,47)(41,48)(42,46);(2,8)(5,9)(6,15)(11,12)(18,21)(19,25)(24,29)(26,28)(33,36,38,46)(34,40,41,37)(35,42,45,44)(39,43,48,47);(1,4,7,3)(2,9,8,5)(6,12,15,11)(10,16,14,13)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,28)(27,32)(33,37)(34,36)(35,43)(38,40)(39,44)(41,46)(42,48)(45,47);(1,5,7,9)(2,4,8,3)(6,13,15,16)(10,12,14,11)(17,19)(18,23)(20,28)(21,27)(22,25)(24,32)(26,30)(29,31)(33,36)(34,40)(35,44)(37,41)(38,46)(39,47)(42,45)(43,48);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,14)(9,16)(17,21,22,18)(19,27,25,23)(20,29,30,24)(26,32,28,31)(33,35)(34,39)(36,44)(37,43)(38,45)(40,47)(41,48)(42,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=g,e^2=g,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, 8851672);