#10419965

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	#57970	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4180 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=g,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10327161);