#9720540

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	#54921	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4149 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,e^2=h,f^2=i,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, 9627736);