#9058501

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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)(3,6)(4,7)(5,10)(8,13)(9,11)(12,15)(14,16)(17,18,23,27)(19,28,31,24)(20,30,29,25)(21,26,32,22);(2,3)(7,9)(8,10)(14,15)(17,19,23,31)(18,24,27,28)(20,32,29,21)(22,30,26,25)(33,34)(35,41)(36,44)(37,38)(39,42)(40,45)(43,47)(46,48);(2,3)(4,11)(5,13)(14,15)(20,29)(21,32)(22,26)(25,30)(34,38)(35,42)(36,45)(47,48);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,20)(18,25)(19,21)(22,28)(23,29)(24,26)(27,30)(31,32)(33,35)(34,39)(36,46)(37,42)(38,41)(40,48)(43,45)(44,47);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,21)(18,26)(19,29)(20,31)(22,27)(23,32)(24,30)(25,28)(33,36)(34,40)(35,43)(37,45)(38,44)(39,47)(41,48)(42,46);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22,23,26)(18,21,27,32)(19,30,31,25)(20,28,29,24)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,45)(46,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^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,d^c=dgi,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, 8965697);