#9243816

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3857 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9151012);