#9015631

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4271 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)(3,8)(4,10)(5,11)(6,16)(7,9)(12,15)(13,14)(17,18,23,26)(19,27,29,24)(20,30,31,21)(22,25,28,32)(33,34)(35,41)(36,44)(37,38)(39,43)(40,46)(42,47)(45,48);(1,3,7,6)(2,8,9,16)(4,12,14,11)(5,10,15,13)(17,19,23,29)(18,24,26,27)(20,25,31,32)(21,28,30,22)(34,38)(39,41)(40,44)(47,48);(2,9)(3,6)(4,14)(5,15)(20,31)(21,30)(22,28)(25,32)(33,35,37,43)(34,39,38,41)(36,45,46,42)(40,48,44,47);(1,4)(2,10)(3,12)(5,16)(6,11)(7,14)(8,15)(9,13)(17,20,23,31)(18,21,26,30)(19,25,29,32)(22,24,28,27)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,48)(45,46);(1,5,7,15)(2,11,9,12)(3,13,6,10)(4,8,14,16)(17,21)(18,20)(19,28)(22,29)(23,30)(24,25)(26,31)(27,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,46)(45,47);(1,6,7,3)(2,8,9,16)(4,11,14,12)(5,10,15,13)(17,22,23,28)(18,25,26,32)(19,21,29,30)(20,24,31,27)(33,36)(34,40)(35,42)(37,46)(38,44)(39,47)(41,48)(43,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=ghi,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, 8922827);