#9020608

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3838 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=gh,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8927804);