#10195821

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

4220 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)(3,9,10,4)(5,13,14,8)(11,12,16,15)(17,18)(19,24)(20,26)(21,27)(22,32)(23,25)(28,29)(30,31)(34,37)(35,41)(40,44)(42,45);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(17,19,23,22)(18,24,25,32)(20,28,27,30)(21,31,26,29)(33,34,38,37)(35,39,41,43)(36,44,46,40)(42,48,45,47);(2,7)(3,10)(8,13)(11,16)(18,25)(20,27)(24,32)(28,30)(33,35,38,41)(34,39,37,43)(36,42,46,45)(40,47,44,48);(1,2)(3,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,20,23,27)(18,26,25,21)(19,28,22,30)(24,29,32,31)(33,35)(34,39)(36,45)(37,43)(38,41)(40,48)(42,46)(44,47);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,21,23,26)(18,27,25,20)(19,29,22,31)(24,28,32,30)(33,36)(34,40)(35,42)(37,44)(38,46)(39,47)(41,45)(43,48);(1,5)(2,8)(3,11)(4,12)(6,14)(7,13)(9,15)(10,16)(17,22,23,19)(18,24,25,32)(20,30,27,28)(21,31,26,29)(33,37,38,34)(35,43,41,39)(36,40,46,44)(42,47,45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=gh,e^2=gh,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, 10103017);