#9713346

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3921 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)(4,5)(7,15)(8,10)(9,16)(11,12)(13,14)(17,18,21,24)(19,22,27,31)(20,23,28,32)(25,29,26,30)(34,39)(35,42)(37,46)(44,47);(1,3,8,5)(2,4,10,6)(7,11,16,13)(9,12,15,14)(17,19)(18,22)(20,26)(21,27)(23,30)(24,31)(25,28)(29,32)(34,39)(36,41)(40,43)(44,47);(1,4,8,6)(2,3,10,5)(7,12,16,14)(9,11,15,13)(17,20)(18,23)(19,25)(21,28)(22,29)(24,32)(26,27)(30,31)(33,34,38,39)(35,43,42,40)(36,37,41,46)(44,48,47,45);(1,5,8,3)(2,4,10,6)(7,13,16,11)(9,12,15,14)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32)(33,35,38,42)(34,40,39,43)(36,44,41,47)(37,45,46,48);(1,6,8,4)(2,3,10,5)(7,12,16,14)(9,13,15,11)(17,20)(18,23)(19,26)(21,28)(22,30)(24,32)(25,27)(29,31)(33,36)(34,37)(35,44)(38,41)(39,46)(40,45)(42,47)(43,48);(1,7)(2,9)(3,11)(4,12)(5,13)(6,14)(8,16)(10,15)(17,19)(18,22)(20,25)(21,27)(23,29)(24,31)(26,28)(30,32)(33,37)(34,41)(35,45)(36,39)(38,46)(40,47)(42,48)(43,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9620542);