#10195483

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	#56349	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3930 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)(3,11)(4,12)(5,9)(6,14)(7,8)(10,13)(15,16)(17,18)(19,24)(20,25)(21,28)(22,23)(26,32)(27,30)(29,31)(33,34,39,37)(35,43,45,40)(36,46,47,41)(38,42,44,48);(1,3,7,12)(2,4,8,11)(5,13,15,6)(9,14,16,10)(17,18)(19,24)(20,29)(21,26)(22,23)(25,31)(27,30)(28,32)(33,35,39,45)(34,40,37,43)(36,44,47,38)(41,48,46,42);(2,8)(4,11)(9,16)(10,14)(18,23)(19,27)(25,29)(26,28)(33,34,39,37)(35,40,45,43)(36,41,47,46)(38,42,44,48);(1,4,7,11)(2,3,8,12)(5,14,15,10)(6,9,13,16)(17,19,22,27)(18,24,23,30)(20,26,31,28)(21,25,32,29)(33,36)(34,41)(35,44)(37,46)(38,45)(39,47)(40,48)(42,43);(1,5,7,15)(2,9,8,16)(3,6,12,13)(4,10,11,14)(17,20)(18,25)(19,28)(21,24)(22,31)(23,29)(26,27)(30,32)(33,37,39,34)(35,43,45,40)(36,41,47,46)(38,42,44,48);(1,6)(2,10)(3,5)(4,9)(7,13)(8,14)(11,16)(12,15)(17,21)(18,26)(19,25)(20,30)(22,32)(23,28)(24,31)(27,29)(33,38,39,44)(34,42,37,48)(35,36,45,47)(40,41,43,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=h,e^2=g,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, 10102679);