#9719146

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3960 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)(4,5)(6,14)(7,9)(10,15)(11,13)(12,16)(17,18)(19,26)(20,24)(21,25)(22,27)(23,30)(28,32)(29,31)(33,34,39,43)(35,42,38,40)(36,37,41,48)(44,45,46,47);(1,3,7,12)(2,8,9,16)(4,13,15,6)(5,11,10,14)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,32)(26,27)(33,35,39,38)(34,40,43,42)(36,44,41,46)(37,47,48,45);(2,9)(3,12)(4,15)(11,14)(17,20,22,21)(18,24,27,25)(19,28,30,29)(23,31,26,32)(33,35,39,38)(34,40,43,42)(36,46,41,44)(37,45,48,47);(1,4)(2,10)(3,13)(5,9)(6,12)(7,15)(8,14)(11,16)(17,21,22,20)(18,25,27,24)(19,29,30,28)(23,32,26,31)(33,36,39,41)(34,37,43,48)(35,44,38,46)(40,47,42,45);(1,5,7,10)(2,4,9,15)(3,14,12,11)(6,16,13,8)(17,19)(18,26)(20,28)(21,29)(22,30)(23,27)(24,32)(25,31)(33,37)(34,41)(35,45)(36,43)(38,47)(39,48)(40,44)(42,46);(1,6,7,13)(2,11,9,14)(3,4,12,15)(5,8,10,16)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32)(33,38,39,35)(34,42,43,40)(36,46,41,44)(37,45,48,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=g,e^2=i,f^2=h,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, 9626342);