#10201913

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	#48972	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3903 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=i,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, 10109109);