#10195907

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	#54723	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3909 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)(3,8)(4,6)(5,10)(7,9)(11,16)(12,15)(13,14)(17,18)(19,25)(20,22)(21,24)(23,27)(26,32)(28,30)(29,31)(33,34,39,37)(35,43,45,40)(36,44,46,41)(38,42,48,47);(1,3)(2,8)(4,13)(5,11)(6,14)(7,12)(9,15)(10,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(33,35)(34,40)(36,42)(37,43)(38,44)(39,45)(41,48)(46,47);(2,9)(3,12)(4,11)(10,14)(18,20)(23,25)(24,28)(31,32)(33,34,39,37)(35,40,45,43)(36,41,46,44)(38,42,48,47);(1,4,7,11)(2,6,9,16)(3,13,12,5)(8,14,15,10)(17,20,22,18)(19,25,27,23)(21,24,30,28)(26,31,29,32)(33,36)(34,41)(35,42)(37,44)(38,43)(39,46)(40,48)(45,47);(1,5,7,13)(2,10,9,14)(3,4,12,11)(6,15,16,8)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,37,39,34)(35,43,45,40)(36,41,46,44)(38,42,48,47);(1,6)(2,11)(3,14)(4,9)(5,8)(7,16)(10,12)(13,15)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(33,38,39,48)(34,42,37,47)(35,44,45,41)(36,43,46,40)

Polycyclic presentation:

2^9|a^2=ghi,c^2=ghi,d^2=h,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, 10103103);