#9015428

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	#55909	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=hi,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, 8922624);