#9938058

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	#53502	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4103 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)(3,7)(4,11)(5,6)(8,13)(9,16)(10,14)(12,15)(17,18)(19,24)(20,21)(22,23)(25,30)(26,27)(28,29)(31,32)(33,34,37,38)(35,39,43,41)(36,44,45,40)(42,47,48,46);(2,6)(7,14)(8,11)(15,16)(18,23)(19,25)(20,28)(27,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(2,6)(7,14)(8,11)(15,16)(34,38)(39,41)(40,44)(46,47);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,19,22,25)(18,24,23,30)(20,26,28,31)(21,27,29,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,20,22,28)(18,21,23,29)(19,26,25,31)(24,27,30,32)(33,35,37,43)(34,39,38,41)(36,42,45,48)(40,46,44,47);(1,4,5,13)(2,8,6,11)(3,9,10,12)(7,15,14,16)(17,21)(18,20)(19,27)(22,29)(23,28)(24,26)(25,32)(30,31)(33,36,37,45)(34,40,38,44)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=i,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9845254);