#8898867

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	#48493	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4089 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)(3,10)(4,9)(5,6)(7,8)(11,16)(12,13)(14,15)(17,18,23,22)(19,25,20,24)(21,26,30,29)(27,32,28,31)(33,34,39,37)(35,42,40,36)(38,45,46,41)(43,44,47,48);(2,8)(5,11)(9,10)(12,14)(17,19,23,20)(18,24,22,25)(21,27,30,28)(26,31,29,32)(34,37)(35,40)(41,45)(43,47);(1,3,7,4)(2,9,8,10)(5,12,11,14)(6,15,16,13)(17,19,23,20)(18,25,22,24)(21,27,30,28)(26,32,29,31)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,48)(46,47);(1,4,7,3)(2,10,8,9)(5,12,11,14)(6,15,16,13)(17,20,23,19)(18,25,22,24)(21,27,30,28)(26,31,29,32)(33,36,39,42)(34,40,37,35)(38,44,46,48)(41,47,45,43);(1,5,7,11)(2,6,8,16)(3,12,4,14)(9,15,10,13)(17,21,23,30)(18,26,22,29)(19,27,20,28)(24,31,25,32)(33,37,39,34)(35,42,40,36)(38,45,46,41)(43,48,47,44);(1,6)(2,11)(3,13)(4,15)(5,8)(7,16)(9,12)(10,14)(17,22,23,18)(19,25,20,24)(21,29,30,26)(27,32,28,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gi,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, 8806063);