#8818460

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3864 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8725656);