#9140545

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	#51926	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4057 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9047741);