#10069121

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	#56378	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3977 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)(17,18,22,26)(19,23,28,32)(20,29,30,24)(21,25,27,31)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,47)(44,48);(1,2,6,4)(3,10,9,7)(5,8,13,12)(11,16,15,14)(17,19,22,28)(18,23,26,32)(20,21,30,27)(24,31,29,25)(33,35)(34,39)(36,38)(37,43)(40,47)(41,42)(44,46)(45,48);(2,4)(7,10)(8,12)(14,16)(17,20,22,30)(18,24,26,29)(19,27,28,21)(23,25,32,31)(33,34,38,41)(35,39,36,42)(37,40,46,45)(43,47,44,48);(1,3,6,9)(2,7,4,10)(5,11,13,15)(8,14,12,16)(17,20)(18,24)(19,27)(21,28)(22,30)(23,25)(26,29)(31,32)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,4,6,2)(3,10,9,7)(5,8,13,12)(11,14,15,16)(17,21,22,27)(18,25,26,31)(19,20,28,30)(23,29,32,24)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,47)(44,48);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,15)(10,16)(17,18)(19,23)(20,24)(21,31)(22,26)(25,27)(28,32)(29,30)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=gh,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, 9976317);