#9814985

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	#46498	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3950 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)(3,7,11,15)(4,9,14,13)(5,12,16,10)(17,18)(19,28)(20,22)(21,24)(23,25)(26,27)(29,32)(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)(17,19,23,21)(18,24,25,28)(20,29,27,31)(22,30,26,32)(33,35,38,40)(34,37,41,42)(36,43,46,47)(39,45,44,48);(2,8)(3,11)(9,13)(10,12)(18,25)(20,27)(24,28)(29,31);(1,3)(2,7)(4,12)(5,13)(6,11)(8,15)(9,16)(10,14)(17,20,23,27)(18,26,25,22)(19,29,21,31)(24,32,28,30)(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)(17,21,23,19)(18,24,25,28)(20,31,27,29)(22,30,26,32)(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)(17,22,23,26)(18,27,25,20)(19,30,21,32)(24,31,28,29)(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=g,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, 9722181);