#9039358

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3905 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)(17,18,22,26)(19,24,29,23)(20,25,31,21)(27,28,30,32)(34,37)(35,41)(40,44)(43,47);(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,19,22,29)(18,23,26,24)(20,27,31,30)(21,28,25,32)(33,34,38,37)(35,42,41,39)(36,44,45,40)(43,46,47,48);(2,6)(4,12)(7,10)(11,15)(17,19,22,29)(18,24,26,23)(20,27,31,30)(21,32,25,28)(33,35,38,41)(34,39,37,42)(36,43,45,47)(40,46,44,48);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20)(18,25)(19,27)(21,26)(22,31)(23,32)(24,28)(29,30)(33,34,38,37)(35,39,41,42)(36,40,45,44)(43,46,47,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19)(18,24)(20,30)(21,28)(22,29)(23,26)(25,32)(27,31)(33,36,38,45)(34,40,37,44)(35,43,41,47)(39,46,42,48);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,21,22,25)(18,20,26,31)(19,28,29,32)(23,27,24,30)(33,37,38,34)(35,42,41,39)(36,44,45,40)(43,48,47,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=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, 8946554);