#8902509

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	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3997 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)(3,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(33,34)(35,42)(36,41)(37,38)(39,40)(43,48)(44,45)(46,47);(1,3)(2,5)(4,11)(6,10)(7,9)(8,14)(12,15)(13,16)(17,18,22,23)(19,21,25,30)(20,24,29,31)(26,28,27,32)(34,40)(37,43)(41,42)(44,46);(2,7)(3,10)(8,15)(11,16)(18,23)(21,30)(24,31)(28,32)(33,35,39,36)(34,41,40,42)(37,44,43,46)(38,47,48,45);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,12)(10,16)(17,19,22,25)(18,21,23,30)(20,27,29,26)(24,32,31,28)(33,36,39,35)(34,42,40,41)(37,44,43,46)(38,47,48,45);(1,5,6,9)(2,3,7,10)(4,12,13,14)(8,16,15,11)(17,20,22,29)(18,24,23,31)(19,26,25,27)(21,28,30,32)(33,37,39,43)(34,38,40,48)(35,44,36,46)(41,47,42,45);(1,3)(2,9)(4,11)(5,7)(6,10)(8,12)(13,16)(14,15)(17,21)(18,25)(19,23)(20,28)(22,30)(24,27)(26,31)(29,32)(33,38)(34,43)(35,45)(36,47)(37,40)(39,48)(41,44)(42,46)

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8809705);