#9040674

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^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, 8947870);