#9719149

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

3803 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=g,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9626345);