#9712481

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	#53893	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3962 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,e^2=g,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, 9619677);