#10166061

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	#38331	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3934 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,d^2=gh,e^2=ghi,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, 10073257);