#10195507

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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=ghi,d^2=h,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, 10102703);