#10166234

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	#48310	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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)(3,8)(4,5)(6,14)(7,9)(10,16)(11,13)(12,15)(17,18)(19,25)(20,21)(22,26)(23,29)(24,30)(27,31)(28,32)(33,34,38,39)(35,43,44,40)(36,41,45,48)(37,46,47,42);(1,3,7,4)(2,8,9,5)(6,11,16,12)(10,15,14,13)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,26)(30,32)(34,39)(36,45)(40,43)(42,46);(2,9)(3,4)(10,14)(11,12)(33,35,38,44)(34,40,39,43)(36,42,45,46)(37,41,47,48);(1,4,7,3)(2,5,9,8)(6,12,16,11)(10,13,14,15)(17,20,22,30)(18,24,26,21)(19,27,29,32)(23,31,25,28)(33,35)(34,40)(36,46)(37,41)(38,44)(39,43)(42,45)(47,48);(1,5)(2,4)(3,9)(6,15)(7,8)(10,11)(12,14)(13,16)(17,21,22,24)(18,20,26,30)(19,28,29,31)(23,27,25,32)(33,36)(34,41)(35,42)(37,43)(38,45)(39,48)(40,47)(44,46);(1,6,7,16)(2,10,9,14)(3,11,4,12)(5,13,8,15)(17,19)(18,25)(20,27)(21,31)(22,29)(23,26)(24,28)(30,32)(33,37,38,47)(34,42,39,46)(35,41,44,48)(36,40,45,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,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, 10073430);