#10195914

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

4057 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)(3,8)(4,6)(5,10)(7,9)(11,16)(12,15)(13,14)(17,18,21,23)(19,25,27,22)(20,24,30,29)(26,32,28,31)(34,38)(36,44)(39,41)(42,48);(1,3)(2,8)(4,13)(5,11)(6,14)(7,12)(9,15)(10,16)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,30)(29,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(2,9)(3,12)(4,11)(10,14)(18,23)(22,25)(24,29)(31,32)(33,34,37,38)(35,39,43,41)(36,40,44,45)(42,46,48,47);(1,4,7,11)(2,6,9,16)(3,13,12,5)(8,14,15,10)(17,18)(19,22)(20,29)(21,23)(24,30)(25,27)(26,32)(28,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,5,7,13)(2,10,9,14)(3,4,12,11)(6,15,16,8)(17,20)(18,24)(19,26)(21,30)(22,31)(23,29)(25,32)(27,28)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,6)(2,11)(3,14)(4,9)(5,8)(7,16)(10,12)(13,15)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,30)(29,31)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=gh,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, 10103110);