#9058493

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	#59617	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4070 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)(3,11)(4,9)(5,15)(6,7)(8,12)(10,16)(13,14)(17,18,23,27)(19,28,30,24)(20,29,31,25)(21,22,26,32);(2,7)(3,12)(9,13)(10,15)(17,19,23,30)(18,24,27,28)(20,21,31,26)(22,29,32,25)(33,34)(35,41)(36,44)(37,38)(39,42)(40,45)(43,47)(46,48);(2,7)(3,12)(4,14)(5,16)(20,31)(21,26)(22,32)(25,29)(34,38)(35,42)(36,45)(47,48);(1,3)(2,8)(4,15)(5,13)(6,12)(7,11)(9,16)(10,14)(17,20,23,31)(18,25,27,29)(19,26,30,21)(22,24,32,28)(33,35)(34,39)(36,46)(37,42)(38,41)(40,48)(43,45)(44,47);(1,4,6,14)(2,9,7,13)(3,10,12,15)(5,11,16,8)(17,21,23,26)(18,22,27,32)(19,20,30,31)(24,25,28,29)(33,36)(34,40)(35,43)(37,45)(38,44)(39,47)(41,48)(42,46);(1,5)(2,10)(3,13)(4,11)(6,16)(7,15)(8,14)(9,12)(17,22)(18,26)(19,29)(20,24)(21,27)(23,32)(25,30)(28,31)(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=gh,e^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, 8965689);