#8202311

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

3897 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8109507);