#10168419

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	#50459	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=gi,e^2=ghi,f^2=h,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, 10075615);