#9716311

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

3893 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)(19,21)(20,29)(22,31)(23,24)(25,30)(26,28)(27,32)(36,44)(40,47)(42,48)(45,46);(1,3,6,11)(2,7,8,4)(5,10,15,14)(9,16,13,12)(17,19,23,27)(18,21,24,32)(20,28,30,22)(25,31,29,26)(33,34)(35,37)(36,40)(38,41)(39,43)(42,45)(44,47)(46,48);(2,8)(4,7)(9,13)(12,16)(18,24)(19,27)(20,30)(26,31)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,48)(45,47);(1,4)(2,3)(5,12)(6,7)(8,11)(9,10)(13,14)(15,16)(17,20)(18,25)(19,28)(21,31)(22,27)(23,30)(24,29)(26,32)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,3)(2,4)(5,14)(6,11)(7,8)(9,16)(10,15)(12,13)(17,21)(18,19)(20,31)(22,29)(23,32)(24,27)(25,28)(26,30)(33,37,38,39)(34,35,41,43)(36,45,44,46)(40,42,47,48);(1,5)(2,9)(3,10)(4,12)(6,15)(7,16)(8,13)(11,14)(17,22,23,28)(18,26,24,31)(19,20,27,30)(21,25,32,29)(33,35)(34,37)(36,42)(38,43)(39,41)(40,45)(44,48)(46,47)

Polycyclic presentation:

2^9|b^2=h,e^2=ghi,f^2=gh,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, 9623507);