#8873727

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	#40244	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3981 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=ghi,d^2=i,e^2=i,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, 8780923);