#9910639

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	#46035	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3890 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9817835);