#9719163

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	#57989	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4158 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:

(1,2,7,9)(3,8,6,16)(4,10,13,5)(11,12,14,15)(17,18)(19,21)(20,29)(22,31)(23,24)(25,30)(26,28)(27,32)(33,34)(35,42)(36,38)(37,41)(39,43)(40,47)(44,48)(45,46);(1,3,7,6)(2,8,9,16)(4,11,13,14)(5,15,10,12)(17,19,23,27)(18,21,24,32)(20,28,30,22)(25,31,29,26)(33,35)(34,40)(36,44)(37,45)(38,46)(39,47)(41,48)(42,43);(2,9)(4,13)(8,16)(11,14)(18,24)(19,27)(20,30)(26,31)(33,36,39,37)(34,38,43,41)(35,44,47,45)(40,46,42,48);(1,4)(2,10)(3,11)(5,9)(6,14)(7,13)(8,12)(15,16)(17,20)(18,25)(19,28)(21,31)(22,27)(23,30)(24,29)(26,32)(33,37,39,36)(34,41,43,38)(35,45,47,44)(40,48,42,46);(1,5,7,10)(2,4,9,13)(3,12,6,15)(8,14,16,11)(17,21)(18,19)(20,31)(22,29)(23,32)(24,27)(25,28)(26,30)(33,35)(34,42)(36,44)(37,45)(38,48)(39,47)(40,43)(41,46);(1,6,7,3)(2,8,9,16)(4,14,13,11)(5,15,10,12)(17,22,23,28)(18,26,24,31)(19,20,27,30)(21,25,32,29)(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=hi,f^2=gh,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, 9626359);