#10147564

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3921 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,23,25)(19,28,22,24)(20,29,30,26)(21,27,32,31)(33,34)(35,41)(36,46)(37,39)(38,42)(40,43)(44,47)(45,48);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,23,22)(18,24,25,28)(20,27,30,31)(21,26,32,29)(33,35,40,37)(34,41,43,39)(36,45,38,44)(42,47,46,48);(2,3)(7,9)(8,10)(14,15)(18,25)(20,30)(24,28)(27,31)(33,36,40,38)(34,42,43,46)(35,44,37,45)(39,47,41,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,30)(18,26,25,29)(19,27,22,31)(21,24,32,28)(33,37,40,35)(34,39,43,41)(36,44,38,45)(42,48,46,47);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21,23,32)(18,27,25,31)(19,29,22,26)(20,28,30,24)(33,38,40,36)(34,42,43,46)(35,45,37,44)(39,47,41,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,19)(18,28,25,24)(20,31,30,27)(21,29,32,26)(33,39)(34,35)(36,47)(37,43)(38,48)(40,41)(42,44)(45,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=h,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, 10054760);