#10195816

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	#56302	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4089 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=h,e^2=gh,f^2=hi,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, 10103012);