#8901444

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

3912 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)(19,26)(20,23)(21,29)(22,25)(24,31)(27,28)(30,32)(33,34,39,43)(35,38,45,40)(36,46,44,41)(37,42,47,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(19,23)(20,26)(27,30)(28,32)(33,35)(34,40)(36,37)(38,43)(39,45)(41,42)(44,47)(46,48);(2,6)(7,13)(8,14)(15,16)(17,19,22,23)(18,20,25,26)(21,30,31,27)(24,32,29,28)(33,36,39,44)(34,41,43,46)(35,37,45,47)(38,48,40,42);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,35)(34,40)(36,37)(38,43)(39,45)(41,42)(44,47)(46,48);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(17,20)(18,23)(19,25)(21,28)(22,26)(24,30)(27,29)(31,32)(33,37)(34,42)(35,44)(36,45)(38,41)(39,47)(40,46)(43,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,21)(18,24)(19,27)(20,28)(22,31)(23,30)(25,29)(26,32)(33,38,39,40)(34,35,43,45)(36,42,44,48)(37,41,47,46)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=ghi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=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, 8808640);