#9016575

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	#55008	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3921 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=ghi,d^2=ghi,e^2=i,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, 8923771);