#9938090

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3878 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,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, 9845286);