#10038558

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3960 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9945754);