#10166165

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	#56730	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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)(3,6,4,8)(5,10,14,16)(11,13,12,15)(17,18)(19,28)(20,31)(21,29)(22,25)(23,27)(24,30)(26,32)(33,34)(35,43)(36,41)(37,42)(38,39)(40,44)(45,48)(46,47);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,19)(18,24)(20,29)(21,25)(22,26)(23,30)(27,28)(31,32)(35,44)(37,45)(40,43)(42,48);(2,9)(3,4)(10,16)(11,12)(17,20,23,22)(18,25,27,31)(19,29,30,26)(21,28,32,24)(33,35,39,44)(34,40,38,43)(36,45,47,37)(41,42,46,48);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,21,23,32)(18,26,27,29)(19,25,30,31)(20,24,22,28)(33,36,39,47)(34,41,38,46)(35,37,44,45)(40,48,43,42);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,22,23,20)(18,25,27,31)(19,29,30,26)(21,24,32,28)(33,37)(34,42)(35,36)(38,48)(39,45)(40,46)(41,43)(44,47);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,19)(18,24)(20,29)(21,25)(22,26)(23,30)(27,28)(31,32)(33,38,39,34)(35,43,44,40)(36,46,47,41)(37,42,45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gi,d^2=h,e^2=hi,f^2=gh,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, 10073361);