#9027829

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	#55213	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3903 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,18,22,25)(19,23,27,21)(20,24,30,32)(26,31,28,29)(33,34,38,42)(35,41,45,39)(36,46,40,37)(43,48,47,44);(1,3,7,4)(2,8,6,9)(5,12,13,11)(10,16,14,15)(17,19)(18,23)(20,26)(21,25)(22,27)(24,31)(28,30)(29,32)(33,35,38,45)(34,39,42,41)(36,47,40,43)(37,44,46,48);(2,6)(3,4)(5,13)(15,16)(17,19,22,27)(18,23,25,21)(20,28,30,26)(24,29,32,31)(33,36,38,40)(34,37,42,46)(35,43,45,47)(39,48,41,44);(1,4,7,3)(2,9,6,8)(5,11,13,12)(10,15,14,16)(17,19)(18,23)(20,28)(21,25)(22,27)(24,29)(26,30)(31,32)(33,37)(34,40)(35,44)(36,42)(38,46)(39,43)(41,47)(45,48);(1,5,7,13)(2,10,6,14)(3,11,4,12)(8,15,9,16)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28)(33,35)(34,41)(36,43)(37,44)(38,45)(39,42)(40,47)(46,48);(1,6,7,2)(3,9,4,8)(5,14,13,10)(11,16,12,15)(17,21,22,23)(18,19,25,27)(20,29,30,31)(24,26,32,28)(33,37)(34,40)(35,44)(36,42)(38,46)(39,43)(41,47)(45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=gh,e^2=gh,f^2=gi,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, 8935025);