#9938074

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4131 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9845270);