#8942665

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	#48651	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

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, 8849861);