#10195618

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	#38868	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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=g,e^2=gh,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, 10102814);