#10480943

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	#57209	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4178 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10388139);