#10195504

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4031 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,d^2=h,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10102700);