#10143267

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3852 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(17,18)(19,24)(20,26)(21,27)(22,29)(23,25)(28,32)(30,31)(33,34,39,38)(35,42,45,40)(36,41,37,46)(43,48,44,47);(2,7)(4,15)(8,11)(10,13)(17,19,23,20)(18,24,25,26)(21,28,31,29)(22,27,32,30)(33,35)(34,40)(36,43)(37,44)(38,42)(39,45)(41,47)(46,48);(2,7)(3,12)(9,14)(10,13)(18,25)(19,20)(27,30)(28,29)(33,36,39,37)(34,41,38,46)(35,43,45,44)(40,47,42,48);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,20,23,19)(18,26,25,24)(21,29,31,28)(22,30,32,27)(33,37,39,36)(34,41,38,46)(35,44,45,43)(40,47,42,48);(1,4)(2,9)(3,13)(5,11)(6,15)(7,14)(8,16)(10,12)(17,21,23,31)(18,27,25,30)(19,29,20,28)(22,26,32,24)(33,38,39,34)(35,42,45,40)(36,46,37,41)(43,48,44,47);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,22,23,32)(18,28,25,29)(19,27,20,30)(21,24,31,26)(33,35)(34,42)(36,43)(37,44)(38,40)(39,45)(41,48)(46,47)

Polycyclic presentation:

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

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, 10050463);