#9206602

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3808 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)(3,9)(4,8)(5,7)(6,11)(10,16)(12,15)(13,14)(19,25)(22,26)(27,31)(29,32)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(1,3)(2,6)(4,12)(5,11)(7,9)(8,16)(10,14)(13,15)(17,18,23,20)(19,22,25,26)(21,28,30,24)(27,32,31,29)(33,35,37,43)(34,38,40,41)(36,44,45,42)(39,47,48,46);(2,7)(6,9)(8,13)(15,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,45)(47,48);(1,2)(3,6)(4,13)(5,7)(8,14)(9,11)(10,16)(12,15)(17,20,23,18)(19,26,25,22)(21,28,30,24)(27,32,31,29)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48);(1,4)(2,8)(3,10)(5,14)(6,15)(7,13)(9,16)(11,12)(17,21,23,30)(18,24,20,28)(19,27,25,31)(22,29,26,32)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47);(1,3)(2,6)(4,12)(5,11)(7,9)(8,16)(10,14)(13,15)(17,22,23,26)(18,25,20,19)(21,29,30,32)(24,31,28,27)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=gh,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=dgi,e^d=egi,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, 9113798);