#10216856

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	#54135	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4073 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)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,18,21,22)(19,23,27,25)(20,28,29,24)(26,31,32,30)(33,34,38,40)(35,42,45,39)(36,44,46,41)(37,47,48,43);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,38,45)(34,39,40,42)(36,43,46,47)(37,44,48,41);(2,6)(7,9)(8,12)(15,16)(18,22)(23,25)(24,28)(30,31)(34,40)(36,46)(39,42)(43,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36,38,46)(34,41,40,44)(35,43,45,47)(37,42,48,39);(1,3)(2,7)(4,13)(5,11)(6,9)(8,16)(10,14)(12,15)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,35)(34,42)(36,47)(37,41)(38,45)(39,40)(43,46)(44,48);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,16)(11,13)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,37)(34,43)(35,44)(36,42)(38,48)(39,46)(40,47)(41,45)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=hi,e^2=ghi,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, 10124052);