#10202031

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	#40773	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3923 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)(3,6)(4,9)(5,10)(7,8)(11,16)(12,14)(13,15)(17,18,23,21)(19,27,29,24)(20,28,30,25)(22,26,32,31)(33,34)(35,40)(36,43)(37,39)(38,45)(41,47)(42,44)(46,48);(1,3,7,11)(2,6,8,16)(4,12,13,5)(9,14,15,10)(17,19)(18,24)(20,26)(21,27)(22,28)(23,29)(25,32)(30,31)(33,35,39,43)(34,36,37,40)(38,42,47,46)(41,44,45,48);(2,8)(3,11)(4,13)(10,14)(17,18,23,21)(19,24,29,27)(20,25,30,28)(22,26,32,31)(33,36,39,40)(34,35,37,43)(38,44,47,48)(41,42,45,46);(1,4)(2,9)(3,12)(5,11)(6,14)(7,13)(8,15)(10,16)(17,20)(18,25)(19,26)(21,28)(22,27)(23,30)(24,32)(29,31)(33,37,39,34)(35,40,43,36)(38,45,47,41)(42,48,46,44);(1,5,7,12)(2,10,8,14)(3,13,11,4)(6,15,16,9)(17,21,23,18)(19,27,29,24)(20,25,30,28)(22,26,32,31)(33,35)(34,40)(36,37)(38,46)(39,43)(41,44)(42,47)(45,48);(1,6)(2,11)(3,8)(4,14)(5,9)(7,16)(10,13)(12,15)(17,22,23,32)(18,26,21,31)(19,28,29,25)(20,27,30,24)(33,38)(34,41)(35,42)(36,44)(37,45)(39,47)(40,48)(43,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=hi,e^2=gh,f^2=ghi,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, 10109227);