#9206418

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9113614);