#10195489

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
prime radical	#41461	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3922 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)(3,9)(4,11)(5,8)(6,7)(10,14)(12,13)(15,16)(17,18)(19,24)(20,25)(21,28)(22,23)(26,32)(27,30)(29,31)(33,34,40,38)(35,44,46,41)(36,42,37,47)(39,43,45,48);(1,3)(2,4)(5,13)(6,11)(7,9)(8,14)(10,15)(12,16)(17,18)(19,24)(20,29)(21,26)(22,23)(25,31)(27,30)(28,32)(33,35)(34,41)(36,39)(37,45)(38,44)(40,46)(42,48)(43,47);(2,7)(4,9)(8,16)(12,14)(18,23)(19,27)(25,29)(26,28)(33,36,40,37)(34,42,38,47)(35,39,46,45)(41,48,44,43);(1,4)(2,3)(5,14)(6,9)(7,11)(8,13)(10,16)(12,15)(17,19,22,27)(18,24,23,30)(20,26,31,28)(21,25,32,29)(33,37,40,36)(34,42,38,47)(35,45,46,39)(41,48,44,43);(1,5,6,15)(2,8,7,16)(3,10,11,13)(4,12,9,14)(17,20)(18,25)(19,28)(21,24)(22,31)(23,29)(26,27)(30,32)(33,38,40,34)(35,44,46,41)(36,47,37,42)(39,43,45,48);(1,3)(2,4)(5,13)(6,11)(7,9)(8,14)(10,15)(12,16)(17,21)(18,26)(19,25)(20,30)(22,32)(23,28)(24,31)(27,29)(33,39,40,45)(34,43,38,48)(35,36,46,37)(41,47,44,42)

Polycyclic presentation:

2^9|a^2=ghi,c^2=ghi,d^2=gh,e^2=g,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, 10102685);