#10195885

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	#46598	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3985 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=gh,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, 10103081);