#9020622

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

3923 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)(19,25)(20,28)(21,22)(23,27)(24,30)(26,31)(29,32)(33,34,39,43)(35,38,37,41)(36,42,47,40)(44,46,45,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,15)(12,13)(14,16)(18,22)(23,25)(24,28)(31,32)(33,35,39,37)(34,38,43,41)(36,45,47,44)(40,46,42,48);(3,10)(4,12)(5,7)(8,16)(17,19,21,27)(18,23,22,25)(20,29,30,26)(24,32,28,31)(33,36,39,47)(34,40,43,42)(35,44,37,45)(38,48,41,46);(1,3)(2,7)(4,13)(5,9)(6,10)(8,14)(11,12)(15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,37,39,35)(34,41,43,38)(36,44,47,45)(40,48,42,46);(1,4)(2,8)(3,11)(5,14)(6,12)(7,15)(9,16)(10,13)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31)(33,36)(34,42)(35,44)(37,45)(38,46)(39,47)(40,43)(41,48);(1,5,6,7)(2,3,9,10)(4,14,12,15)(8,11,16,13)(17,20,21,30)(18,24,22,28)(19,26,27,29)(23,31,25,32)(33,38,39,41)(34,35,43,37)(36,46,47,48)(40,45,42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=gh,f^2=gi,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, 8927818);