#10166718

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	#2673	C₂²
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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=ghi,e^2=gi,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, 10073914);