#9040756

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	#35811	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4044 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=gh,e^2=h,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, 8947952);