#10038507

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	#46723	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3966 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9945703);