#10143769

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	#56350	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)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,18)(19,25)(20,27)(21,24)(22,23)(26,30)(28,29)(31,32)(33,34,40,42)(35,43,39,41)(36,38,37,47)(44,48,45,46);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,19)(18,20)(21,29)(22,27)(23,25)(24,30)(26,31)(28,32)(33,35,40,39)(34,41,42,43)(36,44,37,45)(38,46,47,48);(2,6)(7,9)(8,12)(15,16)(18,23)(20,25)(24,32)(28,30)(33,36,40,37)(34,38,42,47)(35,44,39,45)(41,46,43,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,20)(18,19)(21,30)(22,25)(23,27)(24,29)(26,32)(28,31)(33,37,40,36)(34,38,42,47)(35,45,39,44)(41,46,43,48);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,21,22,31)(18,24,23,32)(19,26,27,29)(20,28,25,30)(33,38)(34,37)(35,46)(36,42)(39,48)(40,47)(41,45)(43,44);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,19)(18,20)(21,29)(22,27)(23,25)(24,30)(26,31)(28,32)(33,39,40,35)(34,41,42,43)(36,45,37,44)(38,46,47,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=ghi,d^2=ghi,e^2=h,f^2=gh,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, 10050965);