#8901349

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3844 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=i,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, 8808545);