#10195416

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4216 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=h,e^2=g,f^2=hi,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, 10102612);