#9029102

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	#43412	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=i,d^2=h,f^2=i,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, 8936298);