#9874916

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3909 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(18,23)(19,26)(25,28)(27,32)(33,34,39,36)(35,40,37,41)(38,42,46,44)(43,47,45,48);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,18,22,23)(19,24,26,21)(20,28,30,25)(27,31,32,29)(33,34,39,36)(35,41,37,40)(38,42,46,44)(43,48,45,47);(2,7)(6,13)(8,11)(14,15)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,29)(28,31)(30,32)(33,36,39,34)(35,41,37,40)(38,44,46,42)(43,48,45,47);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,31,24,29)(33,37,39,35)(34,41,36,40)(38,43,46,45)(42,47,44,48);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,32,28,27)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=i,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9782112);