#8898913

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	#2615	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3998 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)(3,5)(4,9)(6,15)(7,8)(10,14)(11,12)(13,16)(18,19)(21,29)(23,25)(27,32)(33,34,39,44)(35,42,37,40)(36,46,45,41)(38,43,47,48);(1,3,7,6)(2,5,8,15)(4,10,13,12)(9,14,16,11)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,35,39,37)(34,40,44,42)(36,47,45,38)(41,48,46,43);(2,8)(3,6)(9,16)(10,12)(17,18,22,19)(20,23,28,25)(21,26,29,24)(27,31,32,30)(33,36)(34,41)(35,38)(37,47)(39,45)(40,43)(42,48)(44,46);(1,4)(2,9)(3,10)(5,14)(6,12)(7,13)(8,16)(11,15)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31)(33,37,39,35)(34,42,44,40)(36,47,45,38)(41,48,46,43);(1,5)(2,3)(4,11)(6,8)(7,15)(9,12)(10,16)(13,14)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,38,39,47)(34,43,44,48)(35,45,37,36)(40,46,42,41);(1,6,7,3)(2,5,8,15)(4,12,13,10)(9,14,16,11)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=g,e^2=g,f^2=gh,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, 8806109);