#10143704

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

3869 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)(3,10)(4,8)(5,13)(6,7)(9,15)(11,12)(14,16)(17,18,22,24)(19,23,27,29)(20,25,30,32)(21,28,31,26)(33,34)(35,45)(36,42)(37,38)(39,43)(40,44)(41,48)(46,47);(2,7)(4,10)(9,13)(12,14)(17,19)(18,23)(20,26)(21,25)(22,27)(24,29)(28,30)(31,32)(33,35,40,48)(34,41,44,45)(36,46,37,39)(38,47,42,43);(2,7)(3,8)(9,13)(11,16)(18,24)(19,27)(25,32)(26,28)(33,36,40,37)(34,42,44,38)(35,46,48,39)(41,43,45,47);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,14)(15,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,29)(28,30)(31,32)(33,37,40,36)(34,42,44,38)(35,39,48,46)(41,43,45,47);(1,4)(2,8)(3,7)(5,14)(6,10)(9,11)(12,15)(13,16)(17,20,22,30)(18,25,24,32)(19,28,27,26)(21,23,31,29)(33,38,40,42)(34,37,44,36)(35,47,48,43)(39,45,46,41);(1,5,6,15)(2,9,7,13)(3,11,8,16)(4,12,10,14)(17,21,22,31)(18,26,24,28)(19,25,27,32)(20,29,30,23)(33,39)(34,43)(35,36)(37,48)(38,41)(40,46)(42,45)(44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=ghi,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, 10050900);