#9009227

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	#42104	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4075 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)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(33,34,38,42)(35,37,41,39)(36,40,46,44)(43,45,48,47);(2,6)(7,9)(8,12)(14,15)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,30)(28,32)(33,35,38,41)(34,39,42,37)(36,43,46,48)(40,47,44,45);(2,6)(3,10)(4,13)(14,15)(18,22)(19,25)(20,27)(31,32)(33,34,38,42)(35,39,41,37)(36,44,46,40)(43,45,48,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,19,21,25)(18,23,22,29)(20,28,27,26)(24,32,30,31)(33,34)(35,39)(36,44)(37,41)(38,42)(40,46)(43,45)(47,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,20,21,27)(18,24,22,30)(19,26,25,28)(23,31,29,32)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,48)(42,44);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,30)(28,32)(33,37,38,39)(34,41,42,35)(36,45,46,47)(40,48,44,43)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8916423);