#9863149

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

3893 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)(3,7,11,15)(4,9,14,13)(5,12,16,10)(17,18)(19,26)(20,23)(21,25)(22,24)(27,32)(28,29)(30,31)(33,34)(35,42)(36,44)(37,40)(38,41)(39,46)(43,45)(47,48);(1,3)(2,7)(4,12)(5,13)(6,11)(8,15)(9,16)(10,14)(17,19,23,22)(18,24,20,26)(21,27,29,30)(25,31,28,32)(33,35,38,40)(34,37,41,42)(36,43,46,47)(39,45,44,48);(2,8)(3,11)(9,13)(10,12)(18,20)(24,26)(25,28)(31,32);(1,3)(2,7)(4,12)(5,13)(6,11)(8,15)(9,16)(10,14)(17,20,23,18)(19,26,22,24)(21,28,29,25)(27,32,30,31)(33,36)(34,39)(35,43)(37,45)(38,46)(40,47)(41,44)(42,48);(1,4)(2,9)(3,12)(5,15)(6,14)(7,16)(8,13)(10,11)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,37,38,42)(34,40,41,35)(36,45,46,48)(39,47,44,43);(1,5,6,16)(2,10,8,12)(3,13,11,9)(4,7,14,15)(17,22,23,19)(18,26,20,24)(21,27,29,30)(25,31,28,32)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,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, 9770345);