#9020324

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

4237 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,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, 8927520);