#10181174

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	#44426	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10088370);