#10166314

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3926 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)(4,5)(6,10)(7,13)(8,14)(11,15)(12,16)(17,18)(19,24)(20,21)(22,29)(23,25)(26,28)(27,31)(30,32)(33,34,39,44)(35,43,38,40)(36,46,47,41)(37,42,48,45);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19)(18,24)(20,28)(21,26)(22,30)(23,27)(25,31)(29,32)(33,35,39,38)(34,40,44,43)(36,45,47,42)(37,46,48,41);(18,25)(19,27)(20,30)(26,29)(33,36,39,47)(34,41,44,46)(35,45,38,42)(37,43,48,40);(1,4,6,14)(2,8,10,5)(3,11,13,16)(7,15,9,12)(17,20,23,30)(18,21,25,32)(19,28,27,22)(24,26,31,29)(33,36)(34,41)(35,45)(37,40)(38,42)(39,47)(43,48)(44,46);(1,5,6,8)(2,4,10,14)(3,12,13,15)(7,11,9,16)(17,21)(18,20)(19,29)(22,24)(23,32)(25,30)(26,27)(28,31)(33,37)(34,42)(35,41)(36,43)(38,46)(39,48)(40,47)(44,45);(1,3)(2,9)(4,11)(5,15)(6,13)(7,10)(8,12)(14,16)(17,22,23,28)(18,26,25,29)(19,30,27,20)(21,31,32,24)(33,38,39,35)(34,43,44,40)(36,42,47,45)(37,41,48,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=hi,d^2=gh,e^2=ghi,f^2=h,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, 10073510);