#9712803

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	#2699	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3956 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9619999);