#10108662

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	#40982	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4030 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)(3,10)(4,8)(5,13)(6,7)(9,15)(11,12)(14,16)(17,18,23,24)(19,21,26,31)(20,25,29,32)(22,30,28,27)(33,34)(35,41)(36,40)(37,46)(38,42)(39,45)(43,48)(44,47);(2,7)(4,10)(9,13)(12,14)(17,19)(18,21)(20,28)(22,29)(23,26)(24,31)(25,27)(30,32)(33,35)(34,39)(36,43)(37,44)(38,45)(40,47)(41,42)(46,48);(2,7)(3,8)(9,13)(11,16)(18,24)(19,26)(20,29)(27,30)(33,36,38,37)(34,40,42,46)(35,43,45,44)(39,47,41,48);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,14)(15,16)(17,20,23,29)(18,25,24,32)(19,28,26,22)(21,27,31,30)(33,37,38,36)(34,40,42,46)(35,44,45,43)(39,47,41,48);(1,4)(2,8)(3,7)(5,14)(6,10)(9,11)(12,15)(13,16)(17,21)(18,26)(19,24)(20,30)(22,32)(23,31)(25,28)(27,29)(33,34)(35,39)(36,40)(37,46)(38,42)(41,45)(43,47)(44,48);(1,5,6,15)(2,9,7,13)(3,11,8,16)(4,12,10,14)(17,22,23,28)(18,27,24,30)(19,29,26,20)(21,25,31,32)(33,35)(34,41)(36,43)(37,44)(38,45)(39,42)(40,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=gh,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, 10015858);