#10147570

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	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3995 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=gh,e^2=gi,f^2=g,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, 10054766);