#10166113

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	#56414	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4016 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)(3,8)(4,10)(5,11)(6,16)(7,9)(12,13)(14,15)(17,18)(19,27)(20,30)(21,26)(22,24)(23,25)(28,32)(29,31)(33,34,39,44)(35,45,46,40)(36,41,43,38)(37,42,48,47);(1,3,7,6)(2,8,9,16)(4,12,11,14)(5,15,10,13)(17,19,23,22)(18,24,25,27)(20,28,26,31)(21,32,30,29)(35,46)(37,48)(40,45)(42,47);(2,9)(4,11)(8,16)(12,14)(18,25)(20,26)(24,27)(28,31)(33,35,39,46)(34,40,44,45)(36,48,43,37)(38,47,41,42);(1,4,7,11)(2,10,9,5)(3,12,6,14)(8,13,16,15)(17,20)(18,21)(19,28)(22,31)(23,26)(24,32)(25,30)(27,29)(33,36,39,43)(34,41,44,38)(35,37,46,48)(40,47,45,42);(1,5,7,10)(2,11,9,4)(3,13,6,15)(8,12,16,14)(17,21)(18,26)(19,29)(20,25)(22,32)(23,30)(24,28)(27,31)(33,37)(34,42)(35,36)(38,40)(39,48)(41,45)(43,46)(44,47);(1,6,7,3)(2,8,9,16)(4,14,11,12)(5,15,10,13)(17,22,23,19)(18,27,25,24)(20,31,26,28)(21,29,30,32)(33,38,39,41)(34,43,44,36)(35,47,46,42)(37,45,48,40)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=i,d^2=gh,e^2=ghi,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, 10073309);