#9913372

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	#53749	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3879 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)(3,12)(4,5)(6,16)(7,9)(8,13)(10,15)(11,14)(17,18)(19,21)(20,25)(22,26)(23,24)(27,29)(28,30)(31,32)(35,41)(38,45)(42,47)(46,48);(1,3,7,13)(2,8,9,12)(4,14,15,6)(5,16,10,11)(18,24)(21,25)(26,32)(29,30)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(2,9)(3,13)(4,15)(11,16)(17,19,23,20)(18,21,24,25)(22,27,31,28)(26,29,32,30)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4)(2,10)(3,14)(5,9)(6,13)(7,15)(8,11)(12,16)(17,20,23,19)(18,25,24,21)(22,28,31,27)(26,30,32,29)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47);(1,5,7,10)(2,4,9,15)(3,11,13,16)(6,12,14,8)(17,21,23,25)(18,19,24,20)(22,30,31,29)(26,28,32,27)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(1,6,7,14)(2,11,9,16)(3,4,13,15)(5,12,10,8)(17,22,23,31)(18,26,24,32)(19,27,20,28)(21,29,25,30)(33,36)(34,39)(35,42)(37,44)(38,46)(40,43)(41,47)(45,48)

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820568);