#10038431

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	#50736	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4023 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)(19,25)(23,26)(27,31)(30,32)(33,34,38,43)(35,41,44,39)(36,45,46,40)(37,47,48,42);(2,4)(3,9)(8,12)(11,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(35,44)(36,46)(39,41)(40,45);(2,4)(3,9)(8,12)(11,15)(18,22)(23,26)(24,28)(30,32)(33,35,38,44)(34,39,43,41)(36,47,46,42)(37,40,48,45);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32)(33,36,38,46)(34,40,43,45)(35,42,44,47)(37,41,48,39);(1,4,6,2)(3,10,9,7)(5,8,13,12)(11,14,15,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(33,35)(34,41)(36,42)(37,45)(38,44)(39,43)(40,48)(46,47);(1,5,6,13)(2,8,4,12)(3,11,9,15)(7,14,10,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,37,38,48)(34,42,43,47)(35,40,44,45)(36,41,46,39)

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=i,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, 9945627);