#8458461

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

4166 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,e^2=i,f^2=i,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, 8365657);