#10166244

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3945 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)(3,8)(4,5)(6,14)(7,9)(10,16)(11,13)(12,15)(17,18,23,25)(19,28,30,24)(20,31,27,22)(21,26,32,29)(33,34)(35,44)(36,41)(37,48)(38,42)(39,43)(40,47)(45,46);(1,3,7,4)(2,8,9,5)(6,11,16,12)(10,15,14,13)(17,19,23,30)(18,24,25,28)(20,29,27,26)(21,31,32,22)(33,35,39,47)(34,40,43,44)(36,45,48,42)(37,46,41,38);(2,9)(3,4)(10,14)(11,12)(18,25)(20,27)(24,28)(26,29)(33,36,39,48)(34,41,43,37)(35,45,47,42)(38,44,46,40);(1,4,7,3)(2,5,9,8)(6,12,16,11)(10,13,14,15)(17,20,23,27)(18,22,25,31)(19,29,30,26)(21,28,32,24)(33,37)(34,36)(35,46)(38,47)(39,41)(40,45)(42,44)(43,48);(1,5)(2,4)(3,9)(6,15)(7,8)(10,11)(12,14)(13,16)(17,21,23,32)(18,26,25,29)(19,22,30,31)(20,24,27,28)(33,34)(35,40)(36,41)(37,48)(38,45)(39,43)(42,46)(44,47);(1,6,7,16)(2,10,9,14)(3,11,4,12)(5,13,8,15)(17,22)(18,27)(19,21)(20,25)(23,31)(24,26)(28,29)(30,32)(33,38,39,46)(34,42,43,45)(35,37,47,41)(36,44,48,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=h,e^2=hi,f^2=gh,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, 10073440);