#10166278

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3852 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)(3,11)(4,14)(5,10)(6,8)(7,9)(12,16)(13,15)(17,18,23,24)(19,21,26,31)(20,25,29,32)(22,30,28,27)(33,34)(35,42)(36,41)(37,38)(39,43)(40,47)(44,46)(45,48);(1,3,7,6)(2,8,9,11)(4,12,10,15)(5,16,14,13)(17,19)(18,21)(20,28)(22,29)(23,26)(24,31)(25,27)(30,32)(33,35)(34,40)(36,44)(37,45)(38,46)(39,47)(41,48)(42,43);(2,9)(4,10)(8,11)(12,15)(18,24)(19,26)(20,29)(27,30)(33,36,39,37)(34,41,43,38)(35,44,47,45)(40,48,42,46);(1,4)(2,5)(3,12)(6,15)(7,10)(8,16)(9,14)(11,13)(17,20,23,29)(18,25,24,32)(19,28,26,22)(21,27,31,30)(33,37,39,36)(34,41,43,38)(35,45,47,44)(40,48,42,46);(1,5)(2,10)(3,13)(4,9)(6,16)(7,14)(8,12)(11,15)(17,21)(18,26)(19,24)(20,30)(22,32)(23,31)(25,28)(27,29)(33,38,39,41)(34,37,43,36)(35,46,47,48)(40,45,42,44);(1,6,7,3)(2,11,9,8)(4,15,10,12)(5,13,14,16)(17,22,23,28)(18,27,24,30)(19,29,26,20)(21,25,31,32)(33,35)(34,42)(36,44)(37,45)(38,48)(39,47)(40,43)(41,46)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=ghi,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, 10073474);