#9937788

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	#58255	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4080 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^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, 9844984);