#10195485

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	#55126	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3994 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=gh,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, 10102681);