#10196089

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	#54183	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4207 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)(3,12)(4,14)(5,11)(6,8)(7,9)(10,15)(13,16)(17,18)(19,24)(20,26)(21,27)(22,31)(23,25)(28,29)(30,32)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(1,3,7,6)(2,8,9,12)(4,13,15,11)(5,10,16,14)(17,19,23,27)(18,24,25,21)(20,29,30,22)(26,28,32,31)(35,41)(38,45)(42,47)(46,48);(2,9)(4,15)(8,12)(11,13)(18,25)(19,27)(20,30)(28,31)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(1,4)(2,10)(3,13)(5,12)(6,11)(7,15)(8,16)(9,14)(17,20)(18,26)(19,29)(21,31)(22,27)(23,30)(24,28)(25,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,5,7,16)(2,11,9,13)(3,14,6,10)(4,8,15,12)(17,21)(18,27)(19,25)(20,28)(22,32)(23,24)(26,29)(30,31)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47);(1,6,7,3)(2,12,9,8)(4,11,15,13)(5,14,16,10)(17,22,23,29)(18,28,25,31)(19,20,27,30)(21,26,24,32)(33,36)(34,39)(35,42)(37,44)(38,46)(40,43)(41,47)(45,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,e^2=g,f^2=gi,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, 10103285);