#9726619

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	#49808	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3950 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)(4,13)(5,10)(6,14)(7,9)(11,16)(12,15)(17,18)(19,27)(20,25)(21,31)(22,24)(23,28)(26,29)(30,32)(33,34)(35,42)(36,38)(37,41)(39,43)(40,47)(44,48)(45,46);(1,3,7,4)(2,8,9,13)(5,14,15,11)(6,12,16,10)(17,19,22,28)(18,23,24,27)(20,29,30,21)(25,31,32,26)(33,35)(34,40)(36,44)(37,45)(38,46)(39,47)(41,48)(42,43);(2,9)(3,4)(10,12)(11,14)(18,24)(23,27)(25,32)(26,31)(33,36,39,37)(34,38,43,41)(35,44,47,45)(40,46,42,48);(1,4,7,3)(2,8,9,13)(5,14,15,11)(6,10,16,12)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,31)(30,32)(33,37,39,36)(34,41,43,38)(35,45,47,44)(40,48,42,46);(1,5,7,15)(2,10,9,12)(3,11,4,14)(6,8,16,13)(17,20,22,30)(18,25,24,32)(19,21,28,29)(23,26,27,31)(33,35)(34,42)(36,44)(37,45)(38,48)(39,47)(40,43)(41,46);(1,6,7,16)(2,11,9,14)(3,12,4,10)(5,13,15,8)(17,21)(18,26)(19,20)(22,29)(23,25)(24,31)(27,32)(28,30)(33,38,39,41)(34,36,43,37)(35,46,47,48)(40,44,42,45)

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=gh,e^2=h,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, 9633815);