#9863263

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×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3897 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9770459);