#10195616

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	#50068	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3907 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,e^2=gh,f^2=ghi,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, 10102812);