#9938087

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	#41315	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4107 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)(20,25)(21,26)(22,23)(27,29)(28,31)(30,32)(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)(19,26)(20,28)(29,30)(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)(17,19,22,26)(18,24,23,21)(20,27,28,32)(25,29,31,30)(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)(18,25,23,31)(19,27,26,32)(21,30,24,29)(33,35,37,43)(34,39,38,41)(36,42,45,48)(40,46,44,47);(1,4)(2,8)(3,9)(5,13)(6,11)(7,15)(10,12)(14,16)(17,21,22,24)(18,26,23,19)(20,29,28,30)(25,27,31,32)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=i,d^2=gh,e^2=g,f^2=gh,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, 9845283);