#9850850

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3880 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(18,23)(19,26)(25,28)(27,32)(33,34,40,37)(35,38,41,36)(39,45,47,42)(43,44,48,46);(4,11)(8,14)(10,15)(12,16)(17,18)(19,24)(20,28)(21,26)(22,23)(25,30)(27,31)(29,32)(33,35)(34,38)(36,37)(39,43)(40,41)(42,46)(44,45)(47,48);(2,7)(3,9)(8,14)(10,15)(33,36,40,38)(34,41,37,35)(39,44,47,46)(42,48,45,43);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,29,28,31)(33,37,40,34)(35,36,41,38)(39,45,47,42)(43,44,48,46);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,31,24,29)(33,38,40,36)(34,41,37,35)(39,44,47,46)(42,43,45,48);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,27,28,32)(33,39)(34,42)(35,43)(36,44)(37,45)(38,46)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=gh,e^2=gi,f^2=g,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, 9758046);