#9206516

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	#49770	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3967 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=gh,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9113712);