#8826273

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	#54149	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4143 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8733469);