#9206628

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	#42448	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3977 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,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, 9113824);