#10412054

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

4231 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10319250);