#10069092

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	#55104	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3948 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=hi,e^2=i,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, 9976288);