#10171851

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	#44912	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3914 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=hi,e^2=i,f^2=g,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, 10079047);