#9236426

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3727 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=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, 9143622);