#9726074

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3865 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)(3,5)(4,11)(6,14)(7,8)(9,16)(10,15)(12,13)(20,28)(24,31)(27,29)(30,32)(33,34,40,37)(35,44,46,41)(36,47,38,42)(39,48,45,43);(1,3,7,4)(2,5,8,11)(6,10,16,12)(9,13,14,15)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,31)(29,30)(33,35)(34,41)(36,45)(37,44)(38,39)(40,46)(42,43)(47,48);(2,8)(3,4)(9,14)(10,12)(17,19,22,21)(18,23,25,26)(20,27,28,29)(24,30,31,32)(33,36,40,38)(34,42,37,47)(35,45,46,39)(41,43,44,48);(1,4,7,3)(2,5,8,11)(6,12,16,10)(9,13,14,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,37,40,34)(35,44,46,41)(36,42,38,47)(39,48,45,43);(1,5)(2,3)(4,8)(6,15)(7,11)(9,12)(10,14)(13,16)(17,21,22,19)(18,23,25,26)(20,29,28,27)(24,30,31,32)(33,38,40,36)(34,42,37,47)(35,45,46,39)(41,48,44,43);(1,6,7,16)(2,9,8,14)(3,10,4,12)(5,13,11,15)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,39,40,45)(34,43,37,48)(35,38,46,36)(41,42,44,47)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9633270);