#10457920

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

4090 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)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(18,23)(19,25)(21,29)(28,31)(33,34,39,44)(35,40,46,48)(36,37,42,41)(38,47,45,43);(1,2,5,8)(3,6,10,14)(4,7,12,16)(9,13,11,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,35,39,46)(34,40,44,48)(36,45,42,38)(37,43,41,47);(17,19)(18,20)(21,27)(22,25)(23,24)(26,28)(29,32)(30,31)(33,36)(34,41)(35,45)(37,44)(38,46)(39,42)(40,47)(43,48);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(33,36)(34,37)(35,45)(38,46)(39,42)(40,43)(41,44)(47,48);(1,3,5,10)(2,6,8,14)(4,11,12,9)(7,15,16,13)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27)(33,37)(34,42)(35,43)(36,44)(38,40)(39,41)(45,48)(46,47);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,38,39,45)(34,43,44,47)(35,36,46,42)(37,40,41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,e^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, 10365116);