#8901747

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	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=i,e^2=i,f^2=g,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, 8808943);