#8375643

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	#34589	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4006 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8282839);