#10142187

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=ghi,d^2=ghi,e^2=gh,f^2=g,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, 10049383);