#10457911

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

4014 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10365107);