#9851055

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	#47081	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3969 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)(3,8)(4,9)(5,13)(6,7)(10,15)(11,12)(14,16)(17,18,22,24)(19,26,21,23)(20,25,28,31)(27,32,29,30)(33,34)(35,40)(36,46)(37,42)(38,43)(39,44)(41,45)(47,48);(1,2)(3,8)(4,9)(5,10)(6,7)(11,14)(12,16)(13,15)(17,19,22,21)(18,23,24,26)(20,27,28,29)(25,30,31,32)(33,34,39,44)(35,40,47,48)(36,41,45,46)(37,38,43,42);(2,7)(3,9)(10,13)(11,16)(18,24)(23,26)(25,31)(30,32)(33,35,39,47)(34,40,44,48)(36,37,45,43)(38,46,42,41);(1,3,6,9)(2,8,7,4)(5,11,15,16)(10,14,13,12)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,31)(29,32)(33,36,39,45)(34,41,44,46)(35,37,47,43)(38,48,42,40);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,13,16)(17,20)(18,25)(19,27)(21,29)(22,28)(23,30)(24,31)(26,32)(33,37)(34,42)(35,45)(36,47)(38,44)(39,43)(40,41)(46,48);(1,5)(2,10)(3,11)(4,12)(6,15)(7,13)(8,14)(9,16)(17,21,22,19)(18,26,24,23)(20,27,28,29)(25,30,31,32)(33,38,39,42)(34,43,44,37)(35,46,47,41)(36,48,45,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,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, 9758251);