#9040725

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	#49907	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4033 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)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,18,22,24)(19,23,27,31)(20,25,29,32)(21,26,28,30);(1,3,6,11)(2,7,5,9)(4,10,12,13)(8,15,14,16)(17,19)(18,23)(20,21)(22,27)(24,31)(25,26)(28,29)(30,32)(33,34)(35,41)(36,44)(37,38)(39,43)(40,45)(42,47)(46,48);(2,5)(4,12)(7,9)(10,13)(18,24)(19,27)(21,28)(25,32)(34,38)(36,45)(39,41)(42,46);(1,3)(2,7)(4,13)(5,9)(6,11)(8,16)(10,12)(14,15)(17,20,22,29)(18,25,24,32)(19,28,27,21)(23,30,31,26)(33,35)(34,39)(36,46)(37,43)(38,41)(40,48)(42,45)(44,47);(1,4,6,12)(2,8,5,14)(3,10,11,13)(7,15,9,16)(17,21,22,28)(18,26,24,30)(19,20,27,29)(23,25,31,32)(33,36)(34,40)(35,42)(37,45)(38,44)(39,47)(41,48)(43,46);(1,5,6,2)(3,9,11,7)(4,14,12,8)(10,16,13,15)(17,18)(19,23)(20,25)(21,26)(22,24)(27,31)(28,30)(29,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,45)(46,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8947921);