#9206576

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	#58079	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3935 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)(3,8)(4,7)(5,9)(6,11)(10,16)(12,15)(13,14)(17,18)(19,26)(20,25)(21,23)(22,24)(27,32)(28,31)(29,30)(33,34,38,39)(35,41,43,36)(37,45,46,40)(42,44,48,47);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,13)(15,16)(17,19,22,21)(18,23,24,26)(20,28,30,27)(25,32,29,31)(34,39)(36,41)(40,45)(44,47);(4,12)(7,15)(10,13)(14,16)(18,24)(23,26)(25,29)(31,32)(33,34)(35,36)(37,40)(38,39)(41,43)(42,44)(45,46)(47,48);(1,4,5,12)(2,7,9,15)(3,10,11,13)(6,14,8,16)(17,18)(19,23)(20,29)(21,26)(22,24)(25,30)(27,32)(28,31)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,3)(2,8)(4,13)(5,11)(6,9)(7,14)(10,12)(15,16)(17,20,22,30)(18,25,24,29)(19,27,21,28)(23,31,26,32)(33,36)(34,35)(37,44)(38,41)(39,43)(40,42)(45,48)(46,47);(1,2)(3,6)(4,7)(5,9)(8,11)(10,14)(12,15)(13,16)(17,21,22,19)(18,26,24,23)(20,27,30,28)(25,31,29,32)(33,37,38,46)(34,40,39,45)(35,42,43,48)(36,44,41,47)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=gh,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=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, 9113772);