#8436650

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4026 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8343846);