#9017085

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	#44260	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4143 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(33,34,39,43)(35,37,44,40)(36,41,46,48)(38,42,47,45);(2,7)(4,15)(8,11)(10,13)(17,18,22,19)(20,25,27,23)(21,24,29,26)(28,31,32,30)(33,35)(34,40)(36,38)(37,43)(39,44)(41,45)(42,48)(46,47);(2,7)(3,12)(4,15)(5,16)(18,19)(20,27)(21,29)(30,31)(33,36,39,46)(34,41,43,48)(35,38,44,47)(37,42,40,45);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,37,39,40)(34,35,43,44)(36,45,46,42)(38,41,47,48);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,36)(34,41)(35,38)(37,42)(39,46)(40,45)(43,48)(44,47);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,38)(34,42)(35,36)(37,41)(39,47)(40,48)(43,45)(44,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=ghi,d^2=gi,e^2=hi,f^2=h,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, 8924281);