#10042591

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3892 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)(19,26)(24,27)(28,31)(29,32)(33,34,39,45)(35,40,44,38)(36,43,47,41)(37,48,46,42);(2,4)(3,9)(8,12)(11,15)(17,18,22,23)(19,27,26,24)(20,21,25,30)(28,32,31,29)(33,35,39,44)(34,40,45,38)(36,37,47,46)(41,42,43,48);(2,4)(3,9)(8,12)(11,15)(18,23)(20,25)(24,27)(28,31)(33,36,39,47)(34,41,45,43)(35,46,44,37)(38,42,40,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19,22,26)(18,24,23,27)(20,28,25,31)(21,29,30,32)(33,37,39,46)(34,42,45,48)(35,36,44,47)(38,43,40,41);(1,4,6,2)(3,10,9,7)(5,8,13,12)(11,14,15,16)(17,20)(18,21)(19,28)(22,25)(23,30)(24,29)(26,31)(27,32)(33,36)(34,43)(35,46)(37,44)(38,48)(39,47)(40,42)(41,45);(1,5,6,13)(2,8,4,12)(3,11,9,15)(7,14,10,16)(17,21)(18,25)(19,29)(20,23)(22,30)(24,31)(26,32)(27,28)(33,38,39,40)(34,44,45,35)(36,42,47,48)(37,43,46,41)

Polycyclic presentation:

2^9|a^2=h,b^2=i,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, 9949787);