#10171779

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

4069 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)(3,5,9,15)(4,8,13,16)(10,14,11,12)(17,18,22,20)(19,23,27,25)(21,28,29,24)(26,31,32,30)(33,34,39,43)(35,38,42,40)(36,37,41,48)(44,46,47,45);(1,3)(2,5)(4,11)(6,9)(7,15)(8,12)(10,13)(14,16)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,35,39,42)(34,40,43,38)(36,44,41,47)(37,45,48,46);(2,7)(3,9)(8,16)(10,11)(18,20)(23,25)(24,28)(30,31)(33,36,39,41)(34,37,43,48)(35,44,42,47)(38,46,40,45);(1,3)(2,5)(4,11)(6,9)(7,15)(8,12)(10,13)(14,16)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30)(33,34)(35,40)(36,37)(38,42)(39,43)(41,48)(44,45)(46,47);(1,4,6,13)(2,8,7,16)(3,10,9,11)(5,14,15,12)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,37)(34,41)(35,45)(36,43)(38,44)(39,48)(40,47)(42,46);(1,5)(2,3)(4,12)(6,15)(7,9)(8,11)(10,16)(13,14)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,40)(34,42,43,35)(36,46,41,45)(37,47,48,44)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=hi,e^2=i,f^2=g,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, 10078975);