#9040479

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	#49995	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)(3,10)(4,8)(5,13)(6,7)(9,14)(11,12)(15,16)(17,18,22,23)(19,26,27,24)(20,25,30,21)(28,32,31,29)(33,34,37,39)(35,38,41,46)(36,40,44,47)(42,45,43,48);(2,7)(3,8)(9,13)(11,15)(18,23)(19,27)(20,30)(29,32)(33,35)(34,38)(36,43)(37,41)(39,46)(40,48)(42,44)(45,47);(2,7)(4,10)(5,14)(11,15)(20,30)(21,25)(28,31)(29,32)(34,39)(35,41)(40,47)(42,43);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,19,22,27)(18,24,23,26)(20,31,30,28)(21,29,25,32)(33,36,37,44)(34,40,39,47)(35,42,41,43)(38,45,46,48);(1,4)(2,8)(3,7)(5,12)(6,10)(9,15)(11,13)(14,16)(17,20)(18,25)(19,28)(21,23)(22,30)(24,29)(26,32)(27,31)(33,35)(34,38)(36,43)(37,41)(39,46)(40,48)(42,44)(45,47);(1,5,6,14)(2,9,7,13)(3,11,8,15)(4,12,10,16)(17,21,22,25)(18,20,23,30)(19,29,27,32)(24,31,26,28)(33,34)(35,38)(36,40)(37,39)(41,46)(42,45)(43,48)(44,47)

Polycyclic presentation:

2^9|a^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8947675);