#9863944

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	#58573	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,d^2=ghi,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, 9771140);