#9851009

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	#46783	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3964 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=gh,e^2=i,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, 9758205);