#9716283

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3905 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)(3,4)(5,13)(6,8)(7,11)(9,15)(10,16)(12,14)(17,18)(19,22)(20,21)(23,24)(25,29)(26,32)(27,30)(28,31)(36,44)(39,46)(43,48)(45,47);(1,3,6,11)(2,7,8,4)(5,10,15,14)(9,16,13,12)(17,19,23,26)(18,22,24,32)(20,27,29,31)(21,30,25,28)(33,34)(35,40)(36,39)(37,41)(38,42)(43,47)(44,46)(45,48);(2,8)(4,7)(9,13)(12,16)(18,24)(19,26)(20,29)(28,30)(33,35)(34,37)(36,43)(38,41)(39,45)(40,42)(44,48)(46,47);(1,4)(2,3)(5,12)(6,7)(8,11)(9,10)(13,14)(15,16)(17,20)(18,25)(19,27)(21,24)(22,28)(23,29)(26,31)(30,32)(33,36)(34,39)(35,43)(37,45)(38,44)(40,47)(41,48)(42,46);(1,3)(2,4)(5,14)(6,11)(7,8)(9,16)(10,15)(12,13)(17,21,23,25)(18,20,24,29)(19,28,26,30)(22,31,32,27)(33,35)(34,40)(36,43)(37,42)(38,41)(39,47)(44,48)(45,46);(1,5)(2,9)(3,10)(4,12)(6,15)(7,16)(8,13)(11,14)(17,22)(18,26)(19,24)(20,28)(21,27)(23,32)(25,31)(29,30)(33,37,38,40)(34,41,42,35)(36,45,44,47)(39,48,46,43)

Polycyclic presentation:

2^9|b^2=h,e^2=gh,f^2=ghi,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, 9623479);